What else is needed to establish this functional series converges to strictly increasing function?

Question

I have a functional series $$ f(z)=\sum_{n=1}^\infty f_n(z) $$ and am trying to prove that $f(z)$ is strictly increasing for all $z\geq 0$. So far I have proven that

1. The series is absolutely and uniformly convergent for all $z\geq 0$.
2. The series is bounded for all $z\geq 0$, i.e. $0\leq f(z)< \infty$.
3. $f_n(z)>0,\quad \forall n\ \ \land \ \ \forall z\geq 0$.
4. $f_n^\prime(z)>0,\quad \forall n\ \ \land \ \ \forall z\geq 0$.

What else do I need to prove $f(z)$ is strictly increasing?

According to Fubini's theorem on differentiation I know that $$ f^\prime(z)=\sum_{n=1}^\infty f_n^\prime(z). $$ Since I have already shown $f_n^\prime(z)>0$ do I need to prove any other facts to conclude $f^\prime(z)>0$ on $z\geq 0$? It would seem that I am done but am posting this as a sanity check.

Answer 1

You are pretty much already done. By uniform convergence, term by term differentiation is legitimate, hence by 4, $f'(z)=\sum_{n=1}^{\infty} f_n'(z) \gt 0$ and therefore $f(z)$ is strictly increasing.