Exercise · ZW-BOLZANO-WEIERSTRASS-THEOREM-007 · ★★☆ Intermediate

In the bisection proof we start with \([-8,8]\) and at each step keep the half with infinitely many terms. How long is the interval after 3 steps?

Choose one option.

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Explanation

Each step halves the length: \(16/2^3=2\). Since the length tends to \(0\), the intervals nest down to a single point.

Every bounded real sequence \((a_n)\) has at least one subsequence \((a_{n_k})\) that converges to a number in the set \(\mathbb R\) of reals.

\[\exists\,(n_k):\ \lim_{k\to\infty}a_{n_k}\in\mathbb R\]