Compactness of graphs lemma

Compactness of graphs lemma: Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces , let $K\subseteq X$ be compact , and let $f:K\to Y$ be continuous . The graph of $f$ is $\Gamma_f=\{(x,f(x))\in X\times Y: x\in K\}.$ Then $\Gamma_f$ is compact in the product metric space $X\times Y$.

This lemma is useful when turning statements about functions into statements about sets, for example in limiting arguments and compactness proofs.

Proof sketch: Define $F:K\to X\times Y$ by $F(x)=(x,f(x))$. The map $F$ is continuous, and $\Gamma_f=F(K)$ is the continuous image of a compact set , hence compact.