Is the following statement true? Closed and bounded ⇒ compact

No. Closed unit ball in ℓ² (or in C[0,1] with sup norm) e₁, e₂, … has no convergent subsequence since ‖eₙ − eₘ‖ = √2. Heine–Borel is ℝⁿ-only.

Counterexample

Is this true?

Closed and bounded ⇒ compact

No — it is false.

The counterexample

Closed unit ball in $\ell^{2} ($ or in C[0,1] with sup norm)

Why it fails

$e_{1}, e_{2}, \dots$ has no convergent subsequence since ‖ $e_{n} - e_{m}$ ‖ $= \sqrt{2}$ . Heine–Borel is $\mathbb{R}^{n}-$ only.

Keep practicing

Pointwise vs uniform

The mode of convergence — or of continuity — was the wrong one.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.