If $\|f\|_p = \infty$ and $\|g\|_p = \infty$, $p >1$, do we have $\|f+g\|_p = \infty$? If $\|{f}\|_p = \infty$ and $\|g\|_p = \infty$, $p >1$, do we have $\|f+g\|_p = \infty?$ Both $f$ and $g$ are assumed to be non-negative. 
I believe this should be true, but how to prove it? My thought is that I should use the inequality $(a+b)^p \geq a^p+b^p$ if both $a, b$ are nonnegative.

Use $$f(x)+\underbrace{g(x)}_{\geq 0}\geq f(x)$$
for all $x$ to get that, since $p>0$,
$$(f(x)+g(x))^p\geq f(x)^p$$
for all $x$, and then conclude that $$\lVert f+g\rVert_p^p \geq \lVert f\rVert_p^p$$
