Why isn't this proof sufficient for showing that $L^\infty$ is a Banach space?

Question

Suppose we have already shown that $\|\cdot\|_\infty$ is a norm. In order for $L^\infty$ to be a Banach space, if $\{f_n\} \subset L^\infty$ is a Cauchy sequence such that $\| f_n - f\|_\infty \rightarrow 0$, then $f \in L^\infty$.

By the assumption of convergence, choose $N$ such that $$\|f_n - f\| < \epsilon, \quad \forall n \geq N$$ and let $\|f_n\| = C$, where we know $C < \infty$ since $f_n \in L^\infty$. Then, $$\|f\|_\infty \leq \|f_n - f\|_\infty + \|f_n\| \leq \epsilon + C < \infty \\ \implies f \in L^\infty$$ I know this proof is incorrect as I did not even use the fact that $\{f_n\}$ is Cauchy. Also looking at other proofs (for example, this one: $L^{\infty}$ is a Banach Space) I can see that I still have a lot to show.

I would like to know why the above proof fails, does it assume something that has to be shown or that is not correct?

Answer 1

For a Cauchy sequence $(f_n)_n$, you should show that

1. There exists a function $f$ such that $(f_n)_n$ converges to $f$.
2. $f$ belongs to $L^\infty$.

You skipped step 1 by assuming from the start that $(f_n)_n$ converges to some $f$. This is something you have to show.