In the Alexandroff, or one-point compactification of $\mathbb{C}$ (by adding a point $\infty$), consider a function $f:\Omega (\subset \mathbb{C}) \to \mathbb{R}$. I was asked to justify the following definition of $\limsup$ by somebody:
$\displaystyle\limsup_{z \to p} f(z) = \displaystyle\lim_{r \to 0^+} (\sup\{ f(z) : z \in \Omega\ \cap\ U(p;r)\})$
Where $U(p;r)$ is defined as $D(p;r)^*$ (punctured disc around $p$ of radius $r$) if $p \in \mathbb{C}$,and as the annulus centered at $0$ and going from radius $\dfrac{1}{r}$ to $\infty$ if $p \notin \mathbb{C}$. The set of annuli, with $\infty$ added, form basic open sets in the one-point compactification.
I have to justify the following definition by showing that the limit on the right side exists. However, I need a clue on how to start, because in many limit existence problems we have shown the underlying subsequence as Cauchy and used completeness to establish a limit, or figured out a candidate limit by hook or crook. I am not able to do either here.
Thanking all in advance.