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A Short Proof of the Bolzano–Weierstrass Theorem Katrina Eidolon ( and Greg Oman (, University of Colorado, Colorado Springs, CO A fundamental tool used in the analysis of the real line is the well-known Bolzano– Weierstrass theorem: Theorem 1 (Bolzano–Weierstrass Theorem, Version 1). Every bounded sequence of real numbers has a convergent subsequence. (This theorem was originally proved by Bolzano in 1817; it was reproved by Weier- strass in the latter half of the 19th century.) To mention but two applications, the theorem can be used to show that if [a, b] is a closed, bounded interval and f : [a, b] → R is continuous, then f is bounded. One may also invoke the result to establish Can- tor's intersection theorem: If {C n | n ∈ N} is a nested sequence of closed, bounded intervals, then there is a real number belonging to every C i . One would be hard-pressed to find a book on elementary real analysis that does not include Theorem 1. Here is a sketch of one of the most popular proofs: Let (x n ) be a bounded sequence of real numbers. Call a member x n of the sequence a peak if x m ≤ x n for every m ≥ n. If (x n ) has only finitely many peaks, then one can show that (x n ) has a monotone increasing subsequence. Otherwise, one can argue that (x n ) has a monotone decreasing subsequence. In any case, there exists a monotone subsequence (x n k ) of (x n ). This (x n k ) converges to its supremum or infimum according to whether it is increasing or decreasing, respectively. Such an approach can be found in [1, 7, 8]. Another well-known proof begins by noting that since (x n ) is bounded, there exist a 0 , b 0 ∈ R such that {x n | n ∈ N} ⊆ [a 0 , b 0 ]. Let c 0 be the midpoint of a 0 and b 0 . Then either there are infinitely many n for which a n ∈ [a 0 , c 0 ] or there are infinitely many n for which a n ∈ [c 0 , b 0 ]; suppose the former holds. Then take the midpoint c 1 of [a 0 , c 0 ] and repeat the argument. Continuing recursively, one obtains a nested sequence of closed, bounded intervals whose lengths tend to zero. By Cantor's inter- section theorem, there is a (unique) real number x ∗ that lies in every interval. It is then straightforward to obtain a subsequence (x n k ) of (x n ) that converges to x ∗ . One can find this proof in [5, 6]. Still other texts state the Bolzano–Weierstrass Theorem in a slightly different form: Theorem 2 (Bolzano–Weierstrass Theorem, Version 2). Every bounded, infi- nite set of real numbers has a limit point. This can also be proved by the bisection method above [3, 4, 9]. Another approach is to use the Heine–Borel theorem, as indicated in an exercise [1, p. 323]. It is easy to deduce either form of the Bolzano–Weierstrass theorem from the other: Here is an outline for proving the first version from the second. Suppose (x n ) is a bounded sequence. If (x n ) has only finitely many distinct terms, then (x n ) has a con- stant subsequence, which trivially converges. Otherwise, {x n | n ∈ N} is infinite; let L be a limit point. It is not difficult to recursively construct a subsequence of (x n ) converging to L . Short proof. The purpose of this Classroom Capsule is to give a short proof of the second version of the Bolzano–Weierstrass theorem. Our proof hinges upon a set- theoretic observation of the German mathematician Paul St ¨ ackel dating back to 1907. MSC: 26A03 288 © THE MATHEMATICAL ASSOCIATION OF AMERICA

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