Important Note (before closing this as duplicate):
I saw multiple solutions to this problem but all of them using balls which I don't want.
I need to prove that:
If $f : D \to \Bbb R$ is a continuous mapping and $D$ is a compact subset of $\Bbb R^2$, then $f$ is uniformly continuous on $D$.
What do I know?
For all $\epsilon>0$ and all $P_0$ in $D$ there exists $\delta>0$ such that for every $P$ in $C$ the relation $d(P,P_0)<\delta$ implies $d(f(P),f(P_0))<\epsilon$.
The set $D$ is closed and bounded.
What do I need to prove?
For all $\epsilon>0$ there exists $\delta>0$ such that for all $P,P_0$ in $D$ the relation $d(P,P_0)<\delta$ implies $d(f(P),f(P_0))<\epsilon$.