Escaping bump counterexample to global uniform convergence
= Escaping bump counterexample to global uniform convergence
For a compactly supported nonzero function $\psi$, the translates $f_n(x)=\psi(x-n)$ converge uniformly to zero on every compact subset of $\mathbb R$, while $\lVert f_n\rVert_\infty=\lVert\psi\rVert_\infty$ prevents uniform convergence on all of $\mathbb R$.