How can I prove that $\max_{\partial K}\varphi$ does exist?

Question

Let $\varphi:\Omega\to[-\infty,+\infty[$, where $\Omega\in\Bbb C$ is a domain and $\varphi$ is upper semicontinous, i.e. $\varphi(z_0)\ge\limsup_{z\to z_0}\varphi(z)\;\;\;\forall z_0\in\Omega$.

How can I prove, given $K\Subset\Omega$, that $\max_{\partial K}\varphi$ exists finite?

I tried several times but no way seems to be good.

Any help? Thanks a lot

Answer 1

Let $z_k$ be a sequence in $K$ such that $d(z_k,\partial K)\rightarrow 0$ and $$\lim \varphi(z_k)= \sup_{\partial K}\varphi$$

Since $K$ is compact a subsequence converges to $z\in \partial K$, after renaming wlog the original sequence, and of course the image sequence converges to the same value $\sup_{\partial K}\varphi$.

Now apply the semicontinuity condition to that sequence to conclude that $$\sup_{\partial K}\varphi \le \varphi(z)< \infty$$

(Since $z\in \partial K$ this implies that the maximum is attained).