Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · Analysis & Linear Algebra › Metric Spaces

“Closed and bounded ⇒ compact” — false

Counterexample: Closed unit ball in ℓ² (or in C[0,1] with sup norm)

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

compactnessmetric spaces

#2 · Analysis & Linear Algebra › Metric Spaces

“Bounded ⇒ totally bounded” — false

Counterexample: ℝ with the discrete metric

Everything is within distance 1, but no finite set of balls of radius ½ covers it.

compactnessmetric spaces

#3 · Complex Analysis, Algebra & Topology › Topology

“Countably compact implies compact” — false

Counterexample locked — unlock with Notes + PYQ

topologycompactness

#4 · Complex Analysis, Algebra & Topology › Topology

“The closed unit ball of a normed space is compact” — false

Counterexample: The unit ball of ℓ²

Riesz: compactness of the ball holds exactly in finite dimensions.

topologycompactness

#5 · Analysis & Linear Algebra › Sequences and Series of Functions

“A uniformly bounded sequence in C[0,1] has a uniformly convergent subsequence” — false

Counterexample: fₙ(x) = xⁿ

Bounded by 1, but the pointwise limit is discontinuous so no subsequence converges uniformly. Equicontinuity is the missing hypothesis.

uniform convergencecompactness