Normed spaces: Sequence of functions/convergence

Question

$(f_n)$is a sequence of functions in $C([0,1])$

How to prove, that if $f_n$ converges to $ f \in C([0, 1]) $ with respect to $||·||_\infty$ then $f_n$ also converges to $f$ with respect to $ ||·||_1$? Well, so that's it: $$ ||f_n-f||_\infty \to 0 \implies ||f_n-f||_1 \to 0 $$ But I don't know to prove this implication.

Answer 1

For any $\varepsilon > 0$ we can find $N$ s.t. $n > N$ implies $\lVert f_n - f \rVert_{sup} < \varepsilon.$ So $\forall x \in [0,1]$ we have $|f_n(x) - f(x)| < \varepsilon$. Therefore $\int_0^1|f_n(x) - f(x)|\, dx < \int_0^1 \varepsilon \,dx = \varepsilon \to 0.$

Answer 2

$$\int_0^1|f_n-f|\leq \|f_n-f\|_\infty\cdot 1.$$