Let $F= \sum_{j=1}^{\infty} f_j$ where $f_j : [a,b] \to \mathbb R$ are continuous functions and the series is uniformly convergent. Show that $F$ is uniformly continuous.
I'm unsure how to prove this, I first showed that $f_j$ is uniformly continuous since it is continuous on a closed, bounded interval, but then can we deduce that $F$ is also uniformly continuous from this?
I would prove first that $F$ is continuous, then only easily deduce it's uniformly continuous on $[a,b]$. For that first part, it's the very important uniform limit theorem, which you can find in any good analysis book. If you just want a hint on how to prove this theorem, consider the Cauchy criterion, and the inequality $$ |f(x) - f(y)| \leq |f(x) - f_{N_\varepsilon}(x)| + |f_{N_\varepsilon}(x) - f_{N_\varepsilon}(y)| + |f(y) - f_{N_\varepsilon}(y)| $$ for some $N_\varepsilon$ large enough, which does not depend on $x$ or $y$ since the convergence is uniform.
Note: You're using series, I'm using sequences. This doesn't change much by using $F_n = \sum_{j=1}^{n}f_j$, noting that $F$ is the uniform limit of that sequence and taking the argument from there.