Proof: Sum of minimum of two functions vs minimum of sum of functions

Question

I wonder whether the following is true:

$$\min\big\{f_1(x),g_1(x)\big\} + \min\big\{f_2(x),g_2(x)\big\}\le \min\big\{f_1(x)+f_2(x),~g_1(x) + g_2(x)\big\}.$$

I already know how to prove that:

$$\min\{f(x)\}+\min\{g(x)\} \le \min\{f(x)+g(x)\},$$

but I cannot proceed with the handling of the minimum of two functions.

Answer 1

Case 1: Assuming $f_1(x) \ge g_1(x)$ and $f_2(x) \ge g_2(x)$: $$g_1(x) + g_2(x) \le g_1(x) + g_2(x)~\checkmark$$ as $f_1(x)+f_2(x) \ge g_1(x)+g_2(x)$.

Case 2: Assuming $f_1(x) \ge g_1(x)$ and $f_2(x) \le g_2(x)$: $$g_1(x) + f_2(x) \le \min\big\{f_1(x)+f_2(x),~g_1(x) + g_2(x)\big\}$$ The right side of the above equation is minimal in the following cases $f_1(x) = g_1(x)$ or $f_2(x) = g_2(x)$. Thus: $$g_1(x) + f_2(x) \le g_1(x) + f_2(x)~\checkmark$$ Cases 3 and 4 are done analogue.