Proof Regarding Series of Functions.

Question

If it's not too much trouble, may I have some help on this question regarding series of functions?

Let $u$ and $v$ be two series of functions on a set $X$ such that $|u| < |v|$ for every $x$ element of $X$. Prove that if $|v|$ converges uniformly on $X$ then $|u|$ converges uniformly on $X$.

Answer 1

You are given that $\sum_{n=1}^\infty v_n(x)$ converges uniformly. Write out the definition of what this means. (At its heart, inside all the "for all"s and "there exists"s, this will be an inequality.)

You are also given that $|u_n(x)| \le |v_n(x)|$ for every $x\in X$. Combine this with the previous inequality to get a new inequality.

Now check back in with the definition of uniform convergence; your new inequality should be extremely relevant....