Counterexample bank

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

False — $\mathbb{R}$ with the discrete metric

False — (0,1) and $\mathbb{R}$

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

False — f(x) = 1/x on (0,1)

False — Harmonic series $\sum1/n$

False — 0 matrix and [[0,1],[0,0]]

False — [[1,1],[0,1]]

False — $a_{n} = (-1)^{n}$

False — $a_{n} = (-1)^{n}, b_{n} = (-1)^{n}$⁺$^{1}$

False — $a_{n} = (-1)^{n}/\sqrt{n}$

False — $f(x) = \sin(x^{2})$

False — 4×4 nilpotent matrices with Jordan blocks of sizes {2, 2} and {2, 1, 1}

False — A = [[0,1],[0,0]], B = [[0,0],[0,1]]

False — A = B = [[0,1],[0,0]]

False — $f(x) = x^{2} \sin(1/x), g(x) = x$ as x → 0

False — f = 1 on $\mathbb{Q} \cap [0,1], -1$ elsewhere

False — $f_{n}(x) = \sin($nx)/n

False — $\sum(-1)^{n}x^{n} = 1/(1 + x) \to 1/2$

False — T(x) = x + 1/x on $[1, \infty)$

False — Graph of sin(1/x) on (0,1]

False — f with a spike of height n and width $2/n^{3}$ at each integer n

False — $f_{n}(x) = x^{n}$

False — f(x,y) = xy$/(x^{2}+y^{2}), f(0,0) = 0$

False — f(x,y) = (eˣ cos y, eˣ sin y) on $\mathbb{R}^{2}$

False — ∩$_{n\ge1}(-1/n$, 1/n) = {0}

False — The Cantor set

False — $f_{n} = n\cdot1_{(0,1/n)}$ on [0,1]

False — $f(x) = 1/\sqrt{x}$

False — The right shift on $\ell^{2}: (x_{1}, x_{2}, \dots) \mapsto (0, x_{1}, x_{2}, \dots)$

False — The rotation [[0,−1],[1,0]]

False — [[1,1],[0,1]]

False — A = diag(−1, −1)

False — Impossible over $\mathbb{R}$ or $\mathbb{C}$ in finite dimensions

False — $A = I_{2}$ and $B = 2I_{2}$