Let $-\infty<a<b<\infty$. Then prove that $C[a,b]$ along with the norm $$\|u\| \equiv \max_{a\leq x\leq b}|u(x)|$$ is a Banach space.
Attempt: Let $(u_n)$ be a Cauchy sequence in $C[a,b] \implies \|u_n-u_m\|<\epsilon~\forall~n\geq N$. This means $$\max_x |u_n(x)-u_m(x)| < \epsilon~\forall~n\geq N$$
For each $x_0 \in [a,b]$, $$|u_n(x_0)-u_m(x_0)|<\epsilon~\forall~n\geq N$$
Define $y_n \equiv u_n(x_0)$. Then the real sequence $(y_n)$ is Cauchy. Since the set of real numbers over the real field is complete, $(y_n)$ converges (call the limit $y$). Now define $u(x_0) \equiv y$. Repeat the same procedure, replacing $x_0$ by all other remaining points in $[a,b]$. That way we would have constructed the function $u$ on $[a,b]$ whose values $u(x)$ are the limit points of sequences $(u_n(x))$. So
$$u_n(x) \to u(x)~~\forall~x \in [a,b]$$
which means $(u_n)$ is uniformly convergent. Is this approach correct or are there any subtleties that I'm missing? Thanks!