Prove that $\inf \left(f(x) +g(x)\right) \ge \inf f(x) + \inf g(x)$.

Question

I am confused on how to prove this. I think I need an $x\in D$ for some $D$, because I've seen a lot of similar proofs do this. Other than that I am lost, any direction would be helpful, thank you.

Answer 1

For ALL $x\in D$, $$\inf_{D} f\leq f(x)\mbox{ and } \inf_{D} g\leq g(x)\Rightarrow \inf_{D} f +\inf_{D} g\leq f(x)+g(x)$$ which implies $$\inf_{D} f +\inf_{D} g\leq \inf_D(f+g)$$ where in the last step we used the property that if $y\geq b$ for all $y\in Y$ then $\inf Y\geq b$ (recall the definition and the main properties of the infimum).