Here is a quotation in the book "C*-algebras and Finite-Dimensional Approximations":
Instead of considering the *-algebra of finitely supported functions from $\Gamma$ to $C(X)$ (C(X) denotes all the continuous functions on compact Hausdorff space $X$), it is often to think of compactly supported functions $X\times\Gamma\rightarrow \mathbb{C}$. That is, since any compact subset of $X\times \Gamma$ is contained in $X\times F$, for some finite subset $F\subset\Gamma$, we can identify $C_{c}(\Gamma, C(X))$ with $C_{c}(X\times\Gamma)-$an element $\sum f_{s}s$ corresponds to $f\in C_{c}(X\times \Gamma)$, where $f(x,s)=f_{s}(x)$.
My question is why does "any compact subset of $X\times \Gamma$ is contained in $X\times F$, for some finite subset $F\subset\Gamma$" holds? How to comprehend it?
Here $\Gamma$ is a discrete group. Then $F\subset\Gamma$ is compact $\Leftrightarrow$ $F$ is finite. Now if $K\subset X\times\Gamma$ is compact that if we let $\pi_2$ be the projection onto the second coordinate then, since this is continuous, we get that $\pi_2(K)\subset F$ with $F$ finite. Thus $K\subset X\times F$.