The Bolzano Weierstrass Theorem states that every bounded sequence of real numbers has a convergent subsequence [1]. It doesn’t matter how strange or random the sequence appears to be, as long as it is bounded then at least one part of it converges.
This theorem, introduced by Karl Weierstrass in about 1840, was the first major result guaranteeing limits of sequences. A set with this property is called sequentially compact (or sometimes just “compact”) [2].
The theorem has many applications in analysis of the real line, including [3]:
- Showing that if [a, b] is a closed, bounded interval and f : [a, b] → ℝ is continuous, then f is bounded,
- Establishing Cantor’s Intersection Theorem: Suppose {Cn: n ∈ ℕ} is a nested sequence of closed, bounded intervals. This theorem tells us there is a real number that belongs to every Ci.
Proof of the Bolzano Weierstrass Theorem
One of the easier proofs [1]: Take the bounded sequence and cut it in half. For example, if the sequence lies in the interval [0, 1], cut it into two halves: [0, ½] and [½ 1]. the Choose one of the halves with an infinite number of elements in it. This interval has a length of ½. Let’s call this I1. Repeat the process to create interval I2 of length ¼. Repeat over and over, creating a set of nested intervals (I1 ⊇I2 ⊇…) with width tending to zero. If you make a subsequence made up of one term from each interval In, that sequence is always a Cauchy sequence and therefore will converge (According to the Cauchy sequence version of the Completeness Axiom).
References
[1] The Bolzano Weierstrass Property and Compactness. Retrieved July 17, 2021 from: http://www.u.arizona.edu/~mwalker/econ519/Econ519LectureNotes/Bolzano-Weierstrass.pdf
[2] 2 The Bolzano Weierstrass Compactness Theorem of 1840.
[3] Oman, G. (2017). A Short Proof of the Bolzano Weierstrass Theorem. Retrieved July 17, 2021 from: https://faculty.uccs.edu/goman/wp-content/uploads/sites/15/2021/02/A-short-proof-of-the-Bolzano-Weierstrass-Theorem.pdf