Uniform convergence of the sequences $\{f_n\}$ in $C([a,b])$

Question

If I have an uniformly convergent sequence $\{f_n\}$ in the space of continuous functions $C([a,b])$ and $\{f_n\}$ converges uniformly to $f$, then how can I prove that $f \in C([a,b])$?

The uniformly convergent sequence $\{f_n\}$ has to have a limit in $C([a,b])$, otherwise it wouldn't even be convergent. $[a,b]$ is obviously bounded interval. The only thing that came to my mind is to show that $|f_n - f| \lt \epsilon $.

Answer 1

hint

Let $c\in [a,b] $, then for $x\in [a,b] $ and $n\ge 0$,

$$|f (x)-f (c)|\le |f (x)-f_n (x)|+|f_n (x)-f_n (c)|+|f_n (c)-f (c)|$$