In this proof of the completeness of $(C(K), \| \cdot \|_\infty)$ they use the following inequality:
$ \sup_{x \in K} \lim_{m \to \infty} | f_n(x) - f_m(x) | \leq \liminf_{m \to \infty}\, \sup_{x \in K} | f_n(x) - f_m(x) |.$
Can someone explain to me why this is true? Presumably, they first write $ \sup_{x \in K} \lim_{m \to \infty} | f_n(x) - f_m(x) | = \sup_{x \in K} \,\liminf_{m \to \infty} | f_n(x) - f_m(x) |$ and then swap but the exact argument why this gives the inequality is unclear to me.
Many thanks for your help.