I'm trying to understand a step in a proof. I don't get a special trick that is used several times in the book I am reading, so this does not get out of my head. I try to explain the prerequisites and what I don't understand:
Let $Y$ be a Banach space and let $S$ be a set. For a mapping $f:S\to Y$ let $\|f\| := \sup_{x\in S}|f(x)|$. Let $B(S;Y)$ be the set of all mappings $g:S\to Y$ with $\|g\| < \infty$. I don't understand a step in the proof of the fact that this is a complete metric space.
Let $(f_k)_{k\in\mathbb{N}}$ be a Cauchy sequence in $B(S;Y)$. Let $x\in S$. Since $|f_k(x)-f_l(x)| \rightarrow 0$ for $k,l\rightarrow \infty$, we can define the pointwise limit $f$ of the sequence by $f(x) := \lim_{k\to\infty}f_k(x)$. Now here is the step that I don't understand:
The author of the book (and the proof) states that
$\lim_{l\to\infty}|f_l(x)-f_k(x)| \le \lim\inf_{l\to\infty}\|f_l-f_k\|<\infty$.
Why is that? What has the $\lim\inf$ to do here. And why is this finite? I think that it has something to do with subsequences of Cauchy sequences, but I don't understand it.
This $\lim\inf$-trick is used several times in the book so that it seems a bit important to me.
Thank you very much in advance. I'm glad for every help. This does not get out of my head.