I want to prove that $\displaystyle sup \ x \in C[0,1] |f_n(x)-f(x) | \geq ({\displaystyle\int^1_0 |f_n(x)-f(x)|^2 \ dx})^{1/2}$

$\displaystyle ({\displaystyle\int^1_0 |f_n(x)-f(x)|^2 \ dx})^{1/2}$

$\displaystyle \leq$ (by c-s inequality)

$\displaystyle ({\displaystyle\int^1_0 |f_n(x)|^2 \ dx})^{1/2} + ({\displaystyle\int^1_0 |f(x)|^2 \ dx})^{1/2} $

$\displaystyle \leq$

$\displaystyle sup |f_n(x)| + sup|f(x)| $

However I am unsure where to go now