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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?

shansh0201
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  • Probably no: http://math.stackexchange.com/questions/1574573/prove-that-lim-f-nx-n-fx-x-n-rightarrow-x-then-f-n-rightarrow-f-uni?rq=1 – Siminore Nov 15 '16 at 09:32

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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 $0$.

Dominik
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  • Are there any (weak) conditions we can add for it to hold? – MathematicsStudent1122 Nov 15 '16 at 09:38
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    If the domain of $f$ is compact, the theorem holds. I don't think there are any other sensible conditions. For example, uniform boundedness is also too weak, since $f(x) = \min{|x|/n, 1}$ is a conterexample. – Dominik Nov 15 '16 at 09:40