A simplicial graph of $n$ edges is $n$ of copies of interval $[0,1]$ with some of their end points identified. A loop on a simplicial graph with infinitely many edges only passes through the midpoints of finitely many edges.
$f$ is a continuous function on a compact space $X$. $\forall x,y\in X$, let $F(x,y)=f(x)+f(y)-f(x)f(y)$, then $F$ has maximum.
The union of a sequence of compact subsets in a topological space is compact.