Proof of the boundedness theorem by Bolzano–Weierstrass

Builds on Continuous function Bolzano–Weierstrass theorem Bounded sequence Subsequence Limits preserve non-strict inequalities Continuity is equivalent to sequential continuity Sequential continuity Convergent sequences are bounded Archimedean property

Proof

We argue by contradiction.

Let $a \le b$ and let $f \colon [a, b] \to \R$ be continuous. Suppose that $f$ is not bounded.
Then for each natural number $n$ , the number $n$ is not a bound. So there is $x_n \in [a, b]$ with $\abs{f(x_n)} > n$ .
The sequence $(x_n)$ satisfies $a \le x_n \le b$ , so it is bounded. By the Bolzano–Weierstrass theorem, it has a subsequence $(x_{n_k})$ that converges to some real $c$ .
Since $a \le x_{n_k} \le b$ for all $k$ , limits preserve non-strict inequalities, with constant sequences, gives $a \le c \le b$ .
The function $f$ is continuous, so it is sequentially continuous by the equivalence. Hence $f(x_{n_k}) \to f(c)$ .
The convergent sequence $(f(x_{n_k}))$ is bounded: there is $K$ with $\abs{f(x_{n_k})} \le K$ for all $k$ .
The indices satisfy $n_0 < n_1 < \cdots$ . By induction on $k$ , $n_k \ge k$ for all $k$ .
By the Archimedean property, there is a natural number $k$ with $k > K$ .
For this $k$ , steps 2 and 7 give $\abs{f(x_{n_k})} > n_k \ge k > K$ . This contradicts step 6.
So $f$ is bounded: there is $K$ with $\abs{f(x)} \le K$ for all $x \in [a, b]$ .

Idea

If $f$ were unbounded, we could pick points where $\abs{f}$ exceeds $1, 2, 3, \dots$ . These points have a convergent subsequence, and its limit stays in the closed interval. Continuity at that limit would force the values to converge, hence stay bounded, which they do not.

Reading path

To understand this page from the ground up, read these 104 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof of the epsilon-delta form of continuity, Proof that there is a natural number n with 1/n below any positive epsilon, 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 strong induction from ordinary induction, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that limits preserve non-strict inequalities, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity.

Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Integers object
Limit point of a set definition
Mathematical induction concept
Natural numbers object
Ordered field definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Set definition
Ring of integers statement
Limit of a function definition
Archimedean ordered field definition
Successor function 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
Ordered pair definition
Continuity at a point definition
Addition of natural numbers definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
Biconditional definition
The successor function is injective statement
Zero is not a successor statement
Truth table concept
Modus ponens concept
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Completeness axiom statement
Cartesian product definition
Continuous function definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Subset definition
The rationals form an ordered field statement
Propositional logic concept
Mathematical proof concept
Convergent sequence definition
Sequential continuity definition
Binary relation definition
Order on the reals definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Tautology definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Interval definition
Strong induction statement
Logical equivalence definition
Preorder definition
Proof of the epsilon-delta form of continuity proof
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 continuity statement
Well-ordering principle statement
Double negation law statement
Total order definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the triangle inequality for real numbers proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Triangle inequality statement
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof that limits preserve non-strict inequalities proof
The reals are Archimedean statement
Proof that convergent sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Proof that there is a natural number n with 1/n below any positive epsilon proof
Limits preserve non-strict inequalities statement
Convergent sequences are bounded statement
Monotone convergence theorem for sequences statement
There is a natural number n with 1/n below any positive epsilon statement
Proof that a decreasing sequence bounded below converges proof
Decreasing sequence bounded below converges statement
Proof that continuity is equivalent to sequential continuity proof
Continuity is equivalent to sequential continuity statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Bolzano–Weierstrass theorem statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof

Where this leads

1135 pages build on this one.

Tag	`WC9M3CNP02`
URI	`m2://WC9M3CNP02`
Kind	proof
Writing	W0 stub
MSC	26A15
Proves	Boundedness theorem
Uses	Continuous function, Bolzano–Weierstrass theorem, Bounded sequence, Subsequence, Limits preserve non-strict inequalities, Continuity is equivalent to sequential continuity, Sequential continuity, Convergent sequences are bounded, Archimedean property
Changed	4 November 2026

Proof of the boundedness theorem by Bolzano–Weierstrass

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