What is meant by saying that a sequence of functions $f_{n}$ converges uniformly to a function $f$ ?

Let $f_{n}$ be a sequence of differentiable functions on $[a, b]$ with $f_{n}^{\prime}$ continuous and such that $f_{n}\left(x_{0}\right)$ converges for some point $x_{0} \in[a, b]$. Assume in addition that $f_{n}^{\prime}$ converges uniformly on $[a, b]$. Prove that $f_{n}$ converges uniformly to a differentiable function $f$ on $[a, b]$ and $f^{\prime}(x)=\lim _{n \rightarrow \infty} f_{n}^{\prime}(x)$ for all $x \in[a, b]$. [You may assume that the uniform limit of continuous functions is continuous.]

Show that the series

$$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}$$

converges for $s>1$ and is uniformly convergent on $[1+\varepsilon, \infty)$ for any $\varepsilon>0$. Show that $\zeta(s)$ is differentiable on $(1, \infty)$ and

$$\zeta^{\prime}(s)=-\sum_{n=2}^{\infty} \frac{\log n}{n^{s}}$$

[You may use the Weierstrass $M$-test provided it is clearly stated.]