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In my notes, I have written that

\begin{align} \limsup_{x\to a} f(x) & = \inf_{\delta\to0}\sup\left[f(x): \Vert x-a \Vert < \delta \right] \\ & = \inf_{n\to \infty}\sup\left[ f(x): \Vert x-a \Vert <\frac{1}{n} \right] \\ & = \lim_{n\to \infty}\sup\left[ f(x): \Vert x-a\Vert < \frac{1}{n} \right] \end{align}

Can someone help me understand the first line? I thought that the supremum is always a number, so why we are taking the infimum of a number?

Also I tried drawing a picture to illsute what the heck is going on. But basically my gist is that given an interval around $a$, as $\delta \to 0$, that supremum over that interval is the limit supremum

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    See my answer. $ $2012-12-08

1 Answers 1

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For some $a$ and for every $\delta>0$, consider the set $A(\delta,a)=\left\{f(x): 0<\Vert x-a \Vert < \delta \right\}$ and the number $g(\delta,a)=\sup A(\delta,a)$.

Then, $\limsup\limits_{x\to a} f(x)$ is, equivalently, $\inf\limits_{\delta>0}g(\delta,a)$ or $\lim\limits_{\delta\to0}g(\delta,a)$. (But note that $\inf\limits_{\delta\to0}$ does not exist, only $\inf\limits_{\delta>0}$ does.)

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    Well, $B=\{f(x)\mid \|x-a\|\lt\delta\}$ is a set of real numbers and one considers $\inf B$. Do you know the definition of the infimum of a subset of the real line?2012-12-09