I have a question that troubled me for a long time.

If I have two convex discrete function $f(·)$ and $g(·)$ such that $f(·) \ge g(·)$. (may be not necessary?)

Let $x_1 = \text{argmin } f(·)$, and $x_2 = \text{argmin } g(·)$. How to prove that $x_1 \le x_2 $?

Actually, I want to find the upper bound of the minimum $x^*$ of $f(x)$ in order to reduce the enumeration scope to find a global minimum more efficiently.

One possible sufficient condition: if we prove $\Delta f(x) \ge \Delta g(x)$, then $x_1 \le x_2 $. (I hope it's true.)

Why the sufficient condition is correct? Or, there are other approaches to prove $x_1 \le x_2 $?