If $\sum f_j^2 $ and $\sum g_j^2$ conrverge uniformly, show that $\sum f_j g_j$ converges absolutely and uniformly

Question

Let $(f_j)$ and $(g_j)$ be sequences of real functions defined on [a,b]. Assume that both $\Sigma_j f_j^2$ and $\Sigma_j g_j^2$ converge uniformly on $[a,b]$. Show that $\Sigma_j f_j g_j$ converges absolutely and uniformly on $[a,b]$.

My attempt to prove converge absolutely:

We have $(\sum_{j=1}^nf_j g_j)^2\le (\sum_{j=1}^nf_j^2)(\sum_{j=1}^ng_j^2)$

now let $f_j = |f_j| $ and $g_j = |g_j| $, then the above inequality becomes $(\sum_{j=1}^n|f_j g_j|)^2\le (\sum_{j=1}^n|f_j|^2)(\sum_{j=1}^n|g_j|^2)$.

Notice that the RHS, $(\sum_{j=1}^n|f_j|^2)(\sum_{j=1}^n|g_j|^2) = (\sum_{j=1}^nf_j^2)(\sum_{j=1}^ng_j^2)$

Thus, I have $(\sum_{j=1}^n|f_j g_j|)^2\le (\sum_{j=1}^nf_j^2)(\sum_{j=1}^ng_j^2)$.

Because the question said both $\Sigma_j f_j^2$ and $\Sigma_j g_j^2$ converge uniformly on [a,b], that means both series converges on [a,b]. Then I know $(\sum_{j=1}^n|f_j g_j|)^2$ converges?

Answer 1

Hint:
By Cauchy Schwarz inequality, we have
$(\sum_{i=1}^nf_i g_i)^2\le (\sum_{i=1}^nf_i^2)(\sum_{i=1}^ng_i^2)$. Can you finish?