How to prove or disprove:$$\inf_i{\sup_x\{f_i(x)+g_i(x)}\} \leq \inf_i{\sup_x{f_i(x)}} + \inf_i{\sup_x{g_i(x)}}$$,
because,
$f_i(x) \leq \sup\limits_x{f_i(x)}$ and $g_i(x) \leq \sup\limits_x{g_i(x)}$
$\Rightarrow f_i(x) +g_i(x) \leq \sup\limits_x{f_i(x)} + \sup\limits_x{g_i(x)} $
but when I put the infimo I can not get out of it. Help please.
Edit. my space is $\mathbb{R}^3$, I forgot to put information that I think will be important. $f_i, g_i \geq 0$