Subsequences and Bolzano-Weierstrass Theorem: Explained

A subsequence picks out infinitely many terms of a sequence while keeping their order: choose indices n₁ < n₂ < n₃ < … and look at a_{n₁}, a_{n₂}, a_{n₃}, … The Bolzano–Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence.

This matters because it extracts order from chaos: even a wildly oscillating bounded sequence hides a well-behaved convergent subsequence inside. The theorem is the backbone of compactness and of proofs that continuous functions attain maxima on closed intervals.

Key Concepts

• Subsequence notation: (a_{n_k}) with strictly increasing indices n₁ < n₂ < ….
• Bolzano–Weierstrass theorem: bounded ⇒ some subsequence converges.
• Sequential compactness: closed bounded intervals are "compact" because of this theorem.
• Cluster points: limits of convergent subsequences; a divergent bounded sequence can have several.

Formulas

• Subsequence: (a_{n_k}) where n₁ < n₂ < n₃ < … are natural numbers.
• Theorem: if |aₙ| ≤ M for all n, then there exist n_k and L with a_{n_k} → L.
• Bisection idea: repeatedly halve a bounding interval, keeping the half with infinitely many terms.
• Uniqueness link: if every convergent subsequence has the same limit L, the whole sequence converges to L.

Worked Example

Take aₙ = (−1)ⁿ, bounded since |aₙ| = 1. Choose the even indices n_k = 2k: the subsequence is a₂ = 1, a₄ = 1, a₆ = 1, … — constant, hence convergent to 1. Choose the odd indices n_k = 2k − 1: the subsequence is a₁ = −1, a₃ = −1, a₅ = −1, … — convergent to −1. So this one bounded sequence contains two convergent subsequences with different limits, which also proves the original sequence diverges. A richer example: bₙ = sin(n) is bounded by 1 in absolute value, so Bolzano–Weierstrass guarantees a convergent subsequence exists, even though writing one down explicitly is hard. The theorem promises existence without construction.

Common Misconceptions

• The theorem says the sequence converges. It only promises a convergent subsequence — (−1)ⁿ itself diverges.
• Boundedness can be dropped. It cannot: aₙ = n has no convergent subsequence at all.
• The convergent subsequence is easy to find. The proof is non-constructive; for sequences like sin(n) nobody can name the indices.

Real-World Applications

• Proving the Extreme Value Theorem: continuous functions on [a, b] attain their maxima.
• Optimisation: existence of best solutions in bounded feasible sets.
• Signal processing: extracting convergent patterns from bounded noisy data.

Bolzano–Weierstrass guarantees that every bounded sequence hides a convergent subsequence. It turns boundedness — a crude property — into a powerful existence tool for all of analysis.