TITLE: Prove $C[a,b],d_\infty$ is complete.
QUESTION [0 upvotes]: I have this problem with this exercise.

Prove $(C[a,b],d_\infty)$ is complete.

I don't have a very clear idea of how to prove this. But I start to attack the problem in this way:
Let $\{f_n\}$ a Cauchy sequence in $d_\infty$.Then given a $\epsilon >0$ exists a  $N\in \mathbb{N}$ such that if $n,m>N$ then $|f_n(x)-f_m(x)|<\epsilon \ \ \forall x\in [a,b]$.
I'm stuck here, can someone help me? I will be grateful.

REPLY [1 votes]: Hint: for every $x$, $(f_n(x))$ is a Cauchy sequence, call $f(x)$ its limit. Show that $(f_n)$ converges uniformly towards $f$, this implies that $f$ is continuous.
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