Proof that uniform limits of holomorphic functions are holomorphic, by Morera's theorem

Builds on Holomorphic function Heine–Borel theorem Holomorphic functions on a disc have primitives Fundamental theorem of contour integration Estimation lemma Morera's theorem Continuous real function on a compact set attains its bounds Compactness and sequential compactness agree in metric spaces Cauchy's estimates

Proof

Let $U$ , $f_n$ and $f$ be as in the statement. Each $f_n$ is continuous, since a complex differentiable function is continuous at each point.
Claim: $f$ is continuous. Let $z \in U$ and pick $s > 0$ with the closed disc $D = \set{w : \abs{w - z} \le s} \subseteq U$ . It is compact by the Heine–Borel theorem.
Let $\varepsilon > 0$ . By uniform convergence on $D$ , choose $n$ with $\abs{f_n - f} < \varepsilon/3$ on $D$ . Choose $\delta \in (0, s)$ with $\abs{f_n(w) - f_n(z)} < \varepsilon/3$ for $\abs{w - z} < \delta$ .
For $\abs{w - z} < \delta$ , $\abs{f(w) - f(z)} \le \abs{f(w) - f_n(w)} + \abs{f_n(w) - f_n(z)} + \abs{f_n(z) - f(z)} < \varepsilon$ . This proves the claim.
Let $B \subseteq U$ be an open disc, and let $T \subseteq B$ be a closed triangle with boundary path $\partial T$ of length $L$ .
By holomorphic functions on a disc have primitives, each $f_n$ has a primitive on $B$ . By the fundamental theorem of contour integration, $\oint_{\partial T} f_n(z) \, dz = 0$ .
The triangle $T$ is closed and bounded, hence compact. Let $\varepsilon_n = \sup_{T} \abs{f_n - f}$ . Then $\varepsilon_n \to 0$ .
By step 6 and the estimation lemma, $\abs{\oint_{\partial T} f} = \abs{\oint_{\partial T} (f - f_n)} \le \varepsilon_n L$ for each $n$ . So $\oint_{\partial T} f(z) \, dz = 0$ .
By steps 4 and 8 and Morera's theorem on $B$ , $f$ is holomorphic on $B$ .
Each point of $U$ lies in an open disc $B \subseteq U$ . So $f$ is holomorphic on $U$ .
Now let $K \subseteq U$ be compact. If $K$ is empty, there is nothing to prove, so let $K$ be nonempty. Claim: there is $\delta > 0$ such that $K' = \set{w : \abs{w - z} \le \delta \text{ for some } z \in K}$ is a compact subset of $U$ .
If $U = \C$ , take $\delta = 1$ . Otherwise let $d(z) = \inf \set{\abs{z - w} : w \notin U}$ . Then $\abs{d(z) - d(z')} \le \abs{z - z'}$ , so $d$ is continuous.
Since $U$ is open, $d > 0$ on $K$ . By attainment of bounds, $d$ has a positive minimum on $K$ . Let $\delta$ be half of it. Then $K' \subseteq U$ .
The set $K'$ is bounded since $K$ is. It is closed: if $w_j \in K'$ tends to $w$ , pick $z_j \in K$ with $\abs{w_j - z_j} \le \delta$ .
By sequential compactness of $K$ , a subsequence of $(z_j)$ tends to some $z \in K$ . Then $\abs{w - z} \le \delta$ , so $w \in K'$ . By Heine–Borel, $K'$ is compact, which proves the claim.
Let $\eta_n = \sup_{K'} \abs{f_n - f}$ . By uniform convergence on the compact set $K'$ , $\eta_n \to 0$ .
Let $z \in K$ . The closed disc of radius $\delta$ around $z$ lies in $K'$ , and $f_n - f$ is holomorphic on $U$ by step 10.
By Cauchy's estimates with $n = 1$ for $f_n - f$ on this disc, $\abs{f_n'(z) - f'(z)} \le \eta_n / \delta$ .
The bound in step 18 does not depend on $z \in K$ and tends to $0$ . So $f_n' \to f'$ uniformly on $K$ .

Idea

Integrals over triangles pass to the limit under uniform convergence. Each $f_n$ has zero integral over small triangles, so $f$ does too, and Morera's theorem makes $f$ holomorphic. For the derivatives, Cauchy's estimates bound $f_n' - f'$ at a point by the size of $f_n - f$ on a nearby circle.

Reading path

To understand this page from the ground up, read these 376 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of Morera's theorem by a local primitive, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the reverse triangle inequality, Proof of the derivative as a little-o approximation, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that holomorphic functions are infinitely differentiable, Proof that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof of Cauchy's integral formula by a primitive of the difference quotient, Proof of Cauchy's estimates, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the Cauchy–Riemann equations, Proof of the existence of square roots of nonnegative reals, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof that holomorphic functions on a disc have primitives, Proof that compact subsets of a Hausdorff space are closed, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of Cauchy's integral formula for derivatives, Proof of the description of the open sets of a metric space, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof that the modulus is multiplicative, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the estimation lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of integration by substitution, Proof that metric spaces are Hausdorff, Proof of the second fundamental theorem of calculus, Proof that the complex numbers form a field, Proof of the intermediate value theorem by the supremum, Proof that differentiable functions are continuous, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the fundamental theorem of contour integration, Proof of Goursat's theorem by repeated subdivision, Proof of the Archimedean property in an Archimedean ordered field, Proof that a continuous real function on a compact set attains its bounds, Proof of the derivative of a map into a product, Proof that a composition of continuous functions is continuous, Proof of Cantor's intersection theorem, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that holomorphic functions are analytic, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that absolutely convergent series in a Banach space converge, Proof that open balls are open, Proof that power series are holomorphic in the disc of convergence, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that continuous partial derivatives imply differentiability, Proof of the fundamental theorem of calculus, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that sums and products of continuous functions are continuous, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Euclidean metric example
Euclidean space object
Extended real number line object
Function concept
Group concept
Imaginary unit object
Integers object
Limit point of a set definition
Lipschitz continuity definition
Local maximum definition
Mathematical induction concept
Natural numbers object
Norm concept
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Vector space definition
Complex exponential definition
Contour integral definition
Power series in the complex plane definition
Real and imaginary parts definition
Set definition
Fréchet derivative definition
Total derivative concept
Function of several variables concept
Ring definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Archimedean ordered field definition
Successor function definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Norm definition
Circle integral definition
Complex conjugate definition
Disc of convergence definition
Empty set object
Family of sets definition
Ordered pair definition
Limit of a function of several variables concept
Partial derivative concept
Commutative ring definition
Unit of a ring definition
Intersection definition
Union definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Addition of natural numbers definition
Proof of the Archimedean property in an Archimedean ordered field proof
Riemann integrable function definition
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Truth table concept
Modus ponens concept
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Completeness axiom statement
Normed vector space definition
Integral of 1/z around the unit circle example
Proof of the equality of complex numbers proof
Cartesian product definition
Differentiable function of several variables definition
Partial derivative definition
Jacobian matrix concept
Field definition
Continuous function definition
Antiderivative definition
Differentiable function definition
Subset definition
Union of a family of sets definition
One-sided derivative definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Riemann integral definition
The rationals form an ordered field statement
Proof of the derivative as a little-o approximation proof
Propositional logic concept
Upper and lower Darboux sums definition
Mathematical proof concept
Uniqueness quantifier definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Sequential continuity definition
Partial sum definition
Equality of complex numbers statement
Proof of the derivative of a map into R^n by components proof
Binary relation definition
Universal set definition
Topological space definition
Order on the reals definition
Proof that a composition of continuous functions is continuous proof
Derivative function definition
Derivative as a little-o approximation statement
Commutativity of addition statement
Definite integral definition
Proof of strong induction from ordinary induction proof
Upper and lower Darboux integrals definition
Tautology definition
Sum of a finite geometric series statement
Convergent series definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Sequential compactness definition
Interval definition
Complement definition
Higher derivative definition
Proof that the complex numbers form a field proof
Subspace topology definition
Function definition
Strong induction statement
Logical equivalence definition
Sum of a series definition
Absolute convergence definition
Preorder definition
Proof of the epsilon-delta form of the limit of a function proof
Continuously differentiable function definition
Proof of the epsilon-delta form of continuity proof
Image of a set definition
Preimage of a set definition
The complex numbers form a field statement
Metric space definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of a limit statement
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Epsilon-delta definition of continuity statement
Well-ordering principle statement
Diameter of a set definition
Double negation law statement
Total order definition
Bounded set definition
Metric topology definition
Composition of continuous functions is continuous statement
Uniform continuity definition
Proof of the diameter characterisation of Cauchy sequences proof
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Open set definition
Uniform continuity on a set definition
Epsilon characterisation of Cauchy sequences statement
Proof that Lipschitz functions are uniformly continuous proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Hausdorff space definition
Closed set definition
Continuous map of topological spaces definition
Equivalent metrics definition
Proof of the description of the open sets of a metric space proof
Complete metric space definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Open cover definition
Lipschitz functions are uniformly continuous statement
The reals form an ordered field statement
Equivalent norms definition
Open sets of a metric space statement
Open sets characterised by neighbourhoods statement
Compact space definition
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the existence of the infimum by negation proof
Continuous linear map definition
Banach space definition
Proof that closed sets contain their limits proof
Continuity at a point of a topological space definition
Proof that open balls are open proof
Compact set definition
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Closed sets contain their limits statement
Open balls are open statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Triangle inequality statement
Absolute value is multiplicative statement
Square root definition
Proof of the intermediate value theorem by the supremum proof
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Modulus of a complex number definition
Proof of continuity via open sets in metric spaces proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of the reverse triangle inequality proof
Proof that metric spaces are Hausdorff proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof of the limit of a sum of sequences proof
Proof of the uniqueness of limits proof
Compactness via open covers statement
Proof of the squeeze theorem for sequences proof
Proof that limits preserve non-strict inequalities proof
Intermediate value theorem statement
Proof that differentiable functions are continuous proof
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Complex differentiability definition
Proof that a complex number times its conjugate is the square of its modulus proof
Epsilon-delta continuity in metric spaces statement
Proof that compact subsets of a Hausdorff space are closed proof
Reverse triangle inequality statement
Metric spaces are Hausdorff statement
Proof of Cantor's intersection theorem proof
Proof that compact subsets of a metric space are bounded proof
Proof that a continuous image of a compact set is compact proof
Lebesgue number lemma statement
Proof that compactness and sequential compactness agree in metric spaces proof
Proof of the triangle inequality for norms proof
Proof of the sequence characterisation of uniform continuity proof
Differentiable functions are continuous statement
Linearity of the integral statement
Limit of a sum of sequences statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the monotonicity of the integral proof
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Convergent sequences are Cauchy statement
Monotone convergence theorem for sequences statement
Holomorphic function definition
A complex number times its conjugate statement
Continuity via open sets statement
Proof of the Cauchy–Riemann equations proof
Compact subsets of a Hausdorff space are closed statement
Cantor's intersection theorem statement
Compact subsets of a metric space are bounded statement
Continuous image of a compact set is compact statement
Compactness and sequential compactness agree in metric spaces statement
Epsilon-delta uniform continuity statement
Proof of the limit of a product of sequences proof
Monotonicity of the integral statement
There is a natural number n with 1/n below any positive epsilon statement
Darboux and Riemann integrals agree statement
Triangle inequality for norms statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Cauchy–Riemann equations statement
Proof that the modulus is multiplicative proof
Proof that Lipschitz maps are uniformly continuous proof
Proof that absolutely convergent series in a Banach space converge proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Geometric series example
Proof of Riemann's criterion for integrability proof
Limit of a product of sequences statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Decreasing sequence bounded below converges statement
Modulus is multiplicative statement
Absolutely convergent series in a Banach space converge statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Sequential characterization of limits statement
Proof of the ratio test proof
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Bounded linear maps are continuous statement
Proof of the chain rule with an auxiliary difference quotient proof
Comparison test statement
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Proof of the derivative of a map into a product proof
Chain rule statement
Product of continuous functions is continuous statement
Ratio test statement
Additivity of the integral over intervals statement
Proof of the Heine–Borel theorem by sequential compactness proof
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Linearity of differentiation statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Heine–Borel theorem statement
Proof that power series are holomorphic in the disc of convergence proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Boundedness theorem statement
Proof of the constant multiple rule proof
Proof of the first fundamental theorem of calculus proof
Power series are holomorphic in the disc of convergence statement
Derivative of a map into R^n by components statement
Constant multiple rule statement
Proof that a continuous real function on a compact set attains its bounds proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Proof that continuous functions are Riemann integrable proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the estimation lemma proof
Proof of the mean value theorem for integrals proof
Estimation lemma statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that continuous partial derivatives imply differentiability proof
Proof that the integral of a continuous function is an antiderivative proof
Proof of the second fundamental theorem of calculus proof
The integral of a continuous function is an antiderivative statement
Continuous partial derivatives imply differentiability statement
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Proof of integration by substitution proof
Integration by substitution statement
Fundamental theorem of calculus statement
Proof of the fundamental theorem of contour integration proof
Fundamental theorem of contour integration statement
Proof of Goursat's theorem by repeated subdivision proof
Goursat's theorem statement
Proof of Cauchy's integral formula by a primitive of the difference quotient proof
Proof that holomorphic functions on a disc have primitives proof
Cauchy's integral formula statement
Holomorphic functions on a disc have primitives statement
Proof that holomorphic functions are analytic proof
Holomorphic functions are analytic statement
Proof that holomorphic functions are infinitely differentiable proof
Holomorphic functions are infinitely differentiable statement
Proof of Morera's theorem by a local primitive proof
Proof of Cauchy's integral formula for derivatives proof
Cauchy's integral formula for derivatives statement
Morera's theorem statement
Proof of Cauchy's estimates proof
Cauchy's estimates statement
Proof that uniform limits of holomorphic functions are holomorphic, by Morera's theorem proof

Where this leads

3 pages build on this one.

Tag	`R8NWAFFCN8`
URI	`m2://R8NWAFFCN8`
Kind	proof
Writing	W0 stub
MSC	30A99
Proves	Uniform limits of holomorphic functions are holomorphic
Uses	Holomorphic function, Heine–Borel theorem, Holomorphic functions on a disc have primitives, Fundamental theorem of contour integration, Estimation lemma, Morera's theorem, Continuous real function on a compact set attains its bounds, Compactness and sequential compactness agree in metric spaces, Cauchy's estimates
Changed	4 November 2026

Proof that uniform limits of holomorphic functions are holomorphic, by Morera's theorem

Proof

Idea

Reading path

Where this leads

Further reading

What this page needs for W1

What links here