I'm trying to prove the following inequality:
Let $f$ and $g$ be bounded real-valued functions with the same domain. Prove the following:
$\inf(f) + \inf(g) \leqslant \inf(f+g).$
I thought I had proved it, but I made the erroneous assumption that $\inf(f+g)$ can always be expressed in the form $(f+g)(x_1)$ for some $x_1$, which is not necessarily true.