Prove that the sum of the infima is smaller than the infimum of the sum 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.

Another attempt:
Let $a:=\inf (g)$, then $a \le g$, and
$f+a \le f +g$.
Hence
$\inf (f +a) \le \inf (f+g).$
Since $a$ is fixed,
$\inf (f+a)= \inf (f) + a =$
$\inf (f)+ \inf(g) \le \inf (f+g)$.
