Is this correct? Trying to prove that the space $(C(K,\mathbb{R}^m), \| \cdot \|_{\infty})$ with $K\subseteq \mathbb{R}^n$ compact, is complete.

Question

Now we did this proof in lecture and we had proved that if $(f_k)$ was a Cauchy Sequence in the space, then $f_k \to f$ (pointwise) for some function $f: K \rightarrow \mathbb{R}^{m}$.

Then the professor wrote $\|f_{k}-f\|_{\infty} = \lim_{m \to \infty} \|f_k-f_m \|_{\infty}$.

So why is this line correct (if it is)? Haven't we just assumed that the limit of the sup-norm exists which is what we are trying to prove?

Answer 1

He is skipping some details. Here is a proof: given $\epsilon >0$ there exists $p$ such that $\|f_k(x)-f_m(x)\| <\epsilon $ for all $k,m \geq p$ for all $x \in K$. In this inequality we can let $m \to \infty$ to get $\|f_k(x)-f(x)\| \leq \epsilon $ for all $k \geq p$ for all $x \in K$ and this proves thet $f_n \to f$ in the norm.