How to solve the following task:
Show that if $f_n$ is a sequence of uniformly converging mappings $f_n \in C[0,1]$, where $C[0,1]=\{f:[0,1]\rightarrow\mathbb{R} \;\mid\; f\; \text{continuous}\}$ then $$\underset{n\rightarrow \infty}{\lim}\int_0^1 f_n = \int_0^1\underset{n\rightarrow \infty}{\lim} f_n$$
How to tackle this problem? Should I consider using some metric function here or?
Write $$ \left\lvert \int_0^1 f_n - \int_0^1 f\right\rvert \leq \int_0^1 \left\lvert f_n - f\right\rvert \leq \int_0^1 \lVert f_n - f\rVert_\infty = \lVert f_n - f\rVert_\infty. $$ What does uniform convergence give you then?