Let $\{f_n\}$ be a sequence of functions converging pointwise to $f$ such that $\lim_{n \rightarrow \infty } f_n (x_n)=f(x)$ for every sequence $\{x_n \}$ converging to $x$. Then is it true $f_n \rightarrow f$ uniformly?
2026-04-07 08:14:44.1775549684
Let $\{f_n\}$ be a sequence of functions converging pointwise to $f$
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This is in general false. Consider the sequence of functions $f_n(x) = x/n$. Then $f_n(x_n)$ converges to $0$ for every convergent sequence $x_n$, but the sequence $f_n$ does not converge uniformly on $\mathbb{R}$ to $0$.