Question: How to prove that:
$ \sup[f(x)+g(x)]\le \sup(f(x)) + \sup (g(x))$

For me, it sounds very logical and obvious but I did not find the way to prove this. I will be glad for any tips.

Answer: 
Well $\sup(f(x))\geq f(x)$ and $\sup(g(x))\geq g(x)$ by definition of supremum,you can write it like $\sup(f(x))+\sup(g(x))\geq f(x)+g(x)$
**Hint:** You know that $(f+g)(x)=f(x)+g(x)\le \sup f(x)+\sup g(x)$