Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact under the following metrics:
$1. d(f,g)=\sup_{x\in [0,1]}|f(x)-g(x)|$
$2.d(f,g)=\int _0^1 |f(x)-g(x)| dx$
My try:
In order to show not compact if we can find a sequence which has no convergent subsequence then we are done.
How to approach $1,2$?

As mentioned in previous answers, an animation much aids comprehension. Here's an open cover for the closed unit ball with respect to sup norm metric. Let,
\begin{align}
f^n_{\mathrm{upper}}(x) &= \text{the continuous function obtained by connecting the points } \begin{cases} (0,0),\\ (1/n,2),\\(1-1/n,2),\\(1,0) \end{cases}\\
f^n_{\mathrm{lower}}(x) &= \text{the continuous function obtained by connecting the points } \begin{cases} (0,0),\\ (1/n,-2),\\(1-1/n,-2),\\(1,0) \end{cases}. \\
\end{align}
And then define each open set,
$$ \mathcal{O}_n = \{ g(x) \in C[0,1] : f^n_{\mathrm{lower}}(x) < g(x) < f^n_{\mathrm{upper}}(x) \quad \forall  x \in [0,1]\}. $$
And you can check that every function in the unit closed ball lies in at least on of these $ \mathcal{O}_i$s, and each $ \mathcal{O}_i$ is an open set. From the illustation it's clear that the cover does not contain a finite subcover.

The illustration is created using Desmos.
