I'm trying to prove (or disprove) that in the set $C[0,1]$ of continuous (bounded) functions on the real interval [0,1] with the integral norm $\|f(x)\|_1 = \int_0^1|f(x)|dx$ that a sequence of functions $f_n$ is convergent to $f$ if and only if $f_n$ is pointwise-convergent to $f$.
My intuition tells me that this is true, but I'm having trouble formulating an attack (particularly for the convergence => pointwise convergence case). Given $\forall \epsilon > 0 \; \exists N \in \mathbb{N} \;$ $\forall n \geq N \; \int_0^1 |f_n(x) - f(x)|dx < \epsilon $, I'm trying to prove that for an arbitrary $x \in [0,1]$, $\forall \epsilon_0 > 0$ there is an $N_0$ such that for $n_0 \geq N_0 |f_{n_0}(x) - f(x)| < \epsilon_0$. I think I want to make $N_0$ somehow depend on the that integral (or a 'slice' of it), but I don't see it yet. Any pointers/advice/hints are welcome.