Can someone explain how to attack this proofs please. I am studying maths and cannot understand this.
For each $n \in N$, suppose that $f_n \colon [a,b] \to R$ is some function, such that the sequence $\{f_n\}$ converges uniformly to some function $f$. For each $n \in N$, define $F_n \colon [a,b] \to R$ by $$F_n(x)= \int_a^x f_n(t)\,dt.$$ Similarly, define $F \colon [a,b] \to R$ by $$F(x)= \int_a^x f(t)\,dt.$$ Prove that the sequence $\{F_n\}$ converges uniformly to $F$.
Hint:
Note that $$ \left\lvert F_n(x)-F(x)\right\rvert=\left\lvert\int_a^x(f_n(t)-f(t))\,dt\right\rvert\leq\int_a^x\left\lvert f_n(t)-f(t)\right\rvert\,dt\leq\int_a^b\left\lvert f_n(t)-f(t)\right\rvert\,dt $$ for every $x\in[a,b]$. (Why? There are a couple of things to justify there; particularly, the last step.)
Now, you can make $\left\lvert f_n(t)-f(t)\right\rvert$ small uniformly over $t\in[a,b]$ by choosing $n$ sufficiently large. So...