We observe that
$ |f+g|^p \leq (2 \ max \left\{ |f|,|g| \right\} )^p \leq 2^p (|f|^p + |g|^p) $,
which of course implies that
$ ||f+g||_p \leq 2(||f||_p^p + ||g||_p^p)^\frac{1}{p} < \infty $.
If you're short on time during the qual, an alternative (but slightly cheeky, since it's circular) proof is to recall that $ L^p $ spaces are vector spaces.
