So I was looking at a proof that shows that the metric space $(C[0,1],d_\infty)$ is complete.
I know that the definition of complete is that given a metric space X , every cauchy sequence in X converges to a point in X.
The proof was broken into 3 parts
1.create a candidate function and show that the sequence of functions $\{f_n\}$ is cauchy and that the limit $\lim_{n \rightarrow \infty}f_n(a)$ exists.
2.show f is continuous
3.show that $d_\infty(f_n,f)\rightarrow 0$
I understand why we do 1. It's so we can define a cauchy sequence which converges and 3. because we then show that it converges to a point in our space. But I'm not totally sure why we need to show that f is continuous , could someone please explain ?
Note that for completeness you require that your candidate limit $f$ be in $X$. For this, you must show that it is continuous, for otherwise you cannot assert that the limit is in $X = \mathbf C[0,1]$. This is why the second step is important.