Let $(f_n)$ be a sequence of bounded functions on a set $E \subseteq \mathbb R$ and suppose that $f$ is a bounded function such that $\|f_n - f\|_{\infty} \to 0$ as $n \to \infty$. Prove that $(f_n)$ is a Cauchy sequence in the $\sup$ norm.
My Thoughts
Method 1:
Since $(f_n) \to f$ in the $\sup$ norm, we have $$\tag{$*$} \lim_{n \to \infty} \|f_n - f \|_{\infty} = 0 $$ By a previous theorem, we have that $(*)$ is true iff for all $\epsilon >0$ there exists an $N$ so that for all $n \ge N$, $\|f_n - f\|_{\infty} \le \epsilon$.
Now, $$\tag{$\dagger$} \|f_n - f\|_{\infty} = \lim_{m \to \infty} \|f_n - f_m\|_{\infty}$$ My book brings a proof of the converse of this statement to this point and then essentially claims (in more formal language) "since the left hand side of $(\dagger)$ is equal to the right hand side, this implies $$\forall \epsilon > 0 \ \exists M\ \forall n, m > M \Big[\|f_n - f_m\|_{\infty} \le \epsilon\Big]$$
Is this faulty reasoning or not? If so, how else would I prove this?