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