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.

It seems that we can argue in the following simple way and no need to introduce $\varepsilon$: Denote
$a=\inf(f)$ and $b=\inf(g)$. Note that for each $x$ in the domain,
$a\leq f(x)$ and $b\leq g(x)$. Therefore, $a+b\leq f(x)+g(x).$
This shows that $a+b$ is a lower bound of $\{(f+g)(y)\mid y\in\mbox{Domain}\}$
and hence $a+b\leq\inf(f+g)$.
