Counterexample bank

Part C is won by knowing which tempting claims are false. 149 counterexamples; 86 free. The rest come with the Notes pack.

Knowing the claim is false is half of it. The mistakes that cost marks shows the wrong options candidates actually pick, and the reasoning slip behind each one.

free and “A ⇒ B” claims filter by access and by the shape of the statement; everything after them is a subject.

#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.

compactness metric 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.

compactness metric spaces

#4 · Analysis & Linear Algebra › Metric Spaces

“Connected ⇒ path-connected” — false

Counterexample: Topologist's sine curve {(x, sin 1/x) : 0 < x ≤ 1} ∪ {0}×[−1,1]

Connected as the closure of a connected set; no path reaches the segment.

connectedness topology

#5 · Analysis & Linear Algebra › Continuity and Differentiation

“Continuous on a bounded interval ⇒ bounded” — false

Counterexample: f(x) = 1/x on (0,1)

Needs a compact (closed) domain.

continuity

#10 · Analysis & Linear Algebra › The Real Line

“aₙ → 0 ⇒ Σaₙ converges” — false

Counterexample: Harmonic series Σ1/n

series

#14 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Same characteristic polynomial ⇒ similar” — false

Counterexample: 0 matrix and [[0,1],[0,0]]

Both have char poly $x^{2}$ , different minimal polynomials (x vs $x^{2})$ .

linear algebra canonical forms

#19 · Complex Analysis, Algebra & Topology › Groups

“Converse of Lagrange: d | |G| ⇒ subgroup of order d” — false

Counterexample: A₄ has no subgroup of order 6

groups

#28 · Complex Analysis, Algebra & Topology › Singularities and Residues

“|f| bounded near an isolated singularity ⇒ pole” — false

Counterexample: f(z) = sin(z)/z at 0

Bounded ⇒ removable (Riemann). Poles have |f| $\to \infty$ .

complex singularities

#32 · Analysis & Linear Algebra › The Real Line

“Cesàro means converge ⇒ the sequence converges” — false

Counterexample: aₙ = (−1)ⁿ

Partial averages → 0, sequence diverges.

sequences

#35 · Analysis & Linear Algebra › The Real Line

“Σaₙ converges ⇒ Σaₙ² converges” — false

Counterexample: aₙ = (−1)ⁿ/√n

Alternating series converges; squares give the harmonic series.

series

#37 · Analysis & Linear Algebra › Continuity and Differentiation

“Bounded and continuous on ℝ ⇒ uniformly continuous” — false

Counterexample: f(x) = sin(x²)

$x_{n} = 2 π n, y_{n} = 2 π n + π /2$ satisfy | $x_{n} - y_{n}$ | → 0 while | $f (x_{n}) - f (y_{n})$ | = 1.

continuity

#39 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Same characteristic and minimal polynomial ⇒ similar” — false

Counterexample: 4×4 nilpotent matrices with Jordan blocks of sizes {2, 2} and {2, 1, 1}

Both have $χ = x^{4}$ and $m = x^{2}$ , but different numbers of blocks (ranks 2 vs 1).

linear algebra canonical forms

#45 · Analysis & Linear Algebra › Integration

“|f| Riemann integrable ⇒ f Riemann integrable” — false

Counterexample: f = 1 on ℚ ∩ [0,1], −1 elsewhere

|f| ≡ 1 is integrable; f is discontinuous everywhere.

integration

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

“fₙ → f uniformly ⇒ fₙ′ → f′” — false

Counterexample: fₙ(x) = sin(nx)/n

Converges uniformly to 0, derivatives cos(nx) do not converge.

uniform convergence

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

“Σaₙxⁿ → L as x → 1⁻ ⇒ Σaₙ = L” — false

Counterexample: Σ(−1)ⁿxⁿ = 1/(1 + x) → 1/2

$\sum (- 1)^{n}$ diverges. Abel's theorem has no converse without a Tauberian condition.

series power series

#51 · Analysis & Linear Algebra › Metric Spaces

“d(Tx, Ty) < d(x, y) for all x ≠ y on a complete space ⇒ T has a fixed point” — false

Counterexample: T(x) = x + 1/x on [1, ∞)

Distances strictly decrease but no contraction constant k < 1 exists; no fixed point.

completeness fixed point