proof of Weierstrass’ criterion of uniform convergence

The assumption that $|f_n(x)|\le M_n$ for every $x$ guarantees that each numerical series ${\sum }_n{f}_n(x)$ converges absolutely. We call the limit $f(x)$.

To see that the convergence is uniform: let $ϵ>0$. Then there exists $K$ such that $n>K$ implies . Now, if $k>K$,

The $ϵ$ does not depend on $x$, so the convergence is uniform.