Analysis: uniform convergence of a series

Question

Question: Suppose $(a_k)$ and $(b_k)$ are sequences of bounded functions defined on a set S. a is defined and bounded on S, and $a_k\to a$ uniformly on S.

Suppose that the functions $b_k\ge 0$ for all k, and $\sum b_k$ converges uniformly on S. Prove that $\sum a_k b_k$ also converges uniformly on S.

That's the second part of the whole question. I have done the first part, showing that there exists a number M such that $|a_k (x)|\le M$ for all k and all x $\in S$. But I have no idea how it is related to the second part.

Thanks a lot!

Answer 1

There is a version of Weiersrass's "M-test" that solves your problem, that you can find as exercise 9.3.25 at page 383 of "Elementary Real Analysis" by Thomson, Bruckner & Bruckner (2nd edition, 2008). It says that

if $(f_k)_k$ and $(g_k)_k$ are sequences of functions such that $| f_k | \le g_k \ \forall k$, and if $\sum g_k$ converges uniformly, then $\sum f_k$ converges uniformly.

Choose $f_k = a_k b_k$ and $g_k = M b_k$ (where $M$ was obtained in the first part of the question) and you are done.