I'm trying to prove the following inequality:
$``$Let f and g be bounded real-valued functions with the same domain. Prove the following:
$$\inf( f ) + \inf( g ) \le \inf( f+g )"$$
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.