It is well known that limit of function can be converted to limit of sequence. I wonder if it still holds for limit superior of function. This problem is formulated as follows: For function [tex]f:\mathbb R\rightarrow\mathbb R[/tex] and [tex]a\in\mathbb R[/tex], define [tex]{\lim\sup}\limits_{x\to a}f(x)[/tex] to be [tex]\inf\limits_{\delta>0}(\sup\limits_{0<|x-a|<\delta}f(x))[/tex]. Can we have [tex]{\lim\sup}\limits_{x\to a}f(x)=c[/tex] iff [tex]{\lim\sup}\limits_{n\to\infty}f(x_n)=c[/tex] for any sequence [tex]<x_n>[/tex] satisfying 1)[tex]x_n\in\mathbb R[/tex], 2)[tex]x_n\to a[/tex] and 3)[tex]x_n\ne a[/tex]. I have no idea how to prove it, can you help me? Thanks!(adsbygoogle = window.adsbygoogle || []).push({});

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# A question about limit superior for function

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