Do these metrics determine the same Topology. Let $C([0,1])$ denote the set of all continuous real-valued functions on the interval $[0,1]$.
Given elements $f,g ∈ C([0,1])$, define
$d_{\infty}(f,g) = \displaystyle{\text{sup}_{x\in [0,1]}}| f(x)−g(x)|$ and 
$\displaystyle{d_1(f,g) = \int_0^1 | f(x)−g(x)|dx}.$
(a) Show that $d_{\infty}$ is a metric on $C([0,1])$.
(b) Show that $d_1$ is a metric on $C([0,1])$.
(c) Do $d_{\infty}$  and $d_1$ define the same topology on $C([0,1])$?
Part a and part b were fairly easy. For the last part, I know I have to contain one basis element of one metric inside two basis elements of the other metric. My question is, how do you contain a basis element of one metric in the other? 

Hint: For the last part, consider the sequence of functions $f_n$ that are zero everywhere except for a triangular bump going up to $1$ and then back down to zero between the endpoints $1/2^{n+1}$ and $1/2^{n}$. Is this sequence Cauchy in terms of your metric $d_1$, with a limit?. Is the sequence Cauchy according to $d_{\infty}$?
