Statement#

Prove the Bolzano-Weierstrass Theorem:

Every bounded sequence in $\mathbb{R}$ has a convergent subsequence.

Required Topics#

• Bounded sequences
• Subsequences
• Nested interval property
• Monotone subsequences
• Completeness of real numbers

Definitions#

Bounded Sequence: A sequence $\{a_n\}$ is bounded if there exists $M > 0$ such that $|a_n| \leq M$ for all $n$.

Subsequence: A sequence $\{a_{n_k}\}$ where $n_1 < n_2 < n_3 < \cdots$ are indices from $\mathbb{N}$.

Strategy#

Given a bounded sequence $\{a_n\}$:

1. Show that $\{a_n\}$ is contained in some closed interval $[a, b]$
2. Use bisection to construct nested intervals containing infinitely many terms
3. Extract a convergent subsequence using the nested intervals

Solution#

Solution coming soon.