Proof of the isolated zeros theorem by power series

Builds on Holomorphic function Holomorphic functions are analytic Power series in the complex plane Disc of convergence Power series are holomorphic in the disc of convergence

Proof

Let $U$ , $f$ and $z_0$ be as in the statement. Choose $R > 0$ with $D = \set{z : \abs{z - z_0} < R} \subseteq U$ .
By holomorphic functions are analytic, there are $a_0, a_1, \dots \in \C$ with $f(z) = \sum_{n \ge 0} a_n (z - z_0)^n$ for each $z \in D$ .
First, the two cases cannot both hold. If they held with radii $r_1$ and $r_2$ , the point $z = z_0 + \min(r_1, r_2)/2$ would give $f(z) = 0$ and $f(z) \ne 0$ .
If $a_n = 0$ for each $n$ , then $f = 0$ on $D$ , and case 1 holds with $r = R$ .
Otherwise let $m$ be the smallest index with $a_m \ne 0$ . Then $f(z) = \sum_{n \ge m} a_n (z - z_0)^n$ on $D$ .
Consider the power series $\sum_{k \ge 0} a_{m+k} (z - z_0)^k$ . For $z \in D$ with $z \ne z_0$ , its partial sums are $(z - z_0)^{-m}$ times partial sums of the series in step 5. So it converges at each $z \in D$ .
If a power series converges at a point $w$ , its terms tend to $0$ and so are bounded. By the definition of the radius of convergence, the radius is at least $\abs{w - z_0}$ .
By steps 6 and 7, the series of step 6 has radius of convergence at least $R$ . Let $g(z)$ be its sum on $D$ .
By power series are holomorphic, $g$ is holomorphic on $D$ , hence continuous. Also $g(z_0) = a_m \ne 0$ .
By step 6, $f(z) = (z - z_0)^m g(z)$ for each $z \in D$ , including $z = z_0$ where both sides are $0$ if $m \ge 1$ , and $a_0$ if $m = 0$ .
By continuity of $g$ at $z_0$ , there is $r \in (0, R]$ with $g(z) \ne 0$ for $\abs{z - z_0} < r$ .
For $0 < \abs{z - z_0} < r$ , both factors $(z - z_0)^m$ and $g(z)$ are nonzero. So $f(z) \ne 0$ , and case 2 holds.
By steps 3, 4 and 12, exactly one of the two cases holds.

Idea

Near $z_0$ the function is a power series. Either every coefficient is zero, so $f$ vanishes near $z_0$ , or there is a first nonzero coefficient. In the second case $f(z) = (z - z_0)^m g(z)$ with $g(z_0) \ne 0$ , and continuity keeps $g$ away from zero near $z_0$ .

Reading path

To understand this page from the ground up, read these 350 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 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 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 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 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 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 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 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 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 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.

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Proof of the intermediate value theorem by the supremum proof
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Proof that the reals are Archimedean from the completeness axiom proof
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Proof that metric spaces are Hausdorff proof
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 of the Lebesgue number lemma by a finite cover of balls proof
Proof of the limit of a sum of sequences proof
Compactness via open covers statement
Proof of the uniqueness of limits proof
Intermediate value theorem statement
Proof of the squeeze theorem for sequences proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
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Proof that subsequences of a convergent sequence converge to the same limit proof
The reals are Archimedean statement
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Proof that compact subsets of a metric space are bounded proof
Proof that compactness and sequential compactness agree in metric spaces proof
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Proof that a continuous image of a compact set is compact 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
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Proof that there is a natural number n with 1/n below any positive epsilon proof
Uniqueness of limits statement
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Proof of the monotonicity of the integral proof
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
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Compact subsets of a Hausdorff space are closed statement
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Compactness and sequential compactness agree in metric spaces statement
Epsilon-delta uniform continuity statement
Continuous image of a compact set is compact statement
Proof of the limit of a product of sequences proof
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon 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
Cauchy–Riemann equations statement
Proof that a linear map between normed spaces is continuous if and only if it is bounded 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
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
Comparison test statement
Proof of the chain rule with an auxiliary difference quotient proof
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
Product of continuous functions is continuous statement
Chain rule statement
Ratio test statement
Proof of the Heine–Borel theorem by sequential compactness proof
Additivity of the integral over intervals statement
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
Cauchy's integral formula statement
Proof that holomorphic functions are analytic proof
Holomorphic functions are analytic statement
Proof of the isolated zeros theorem by power series proof

Where this leads

51 pages build on this one.

Tag	`QJXMJ95CWK`
URI	`m2://QJXMJ95CWK`
Kind	proof
Writing	W0 stub
MSC	30C15
Proves	Isolated zeros theorem
Uses	Holomorphic function, Holomorphic functions are analytic, Power series in the complex plane, Disc of convergence, Power series are holomorphic in the disc of convergence
Changed	4 November 2026

Proof of the isolated zeros theorem by power series

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