Any ideas to solve this problem?
Let $f\colon \mathbb{R}_{+} \to \mathbb{R}$ uniformly continuous. Prove that exists $K>0$ such that for each $x\in \mathbb{R}_{+},$ $$\sup_{w>0}\{ |f(x+w) -f(w)|\}\le K \,\, ( x + 1).$$
Any ideas to solve this problem?
Let $f\colon \mathbb{R}_{+} \to \mathbb{R}$ uniformly continuous. Prove that exists $K>0$ such that for each $x\in \mathbb{R}_{+},$ $$\sup_{w>0}\{ |f(x+w) -f(w)|\}\le K \,\, ( x + 1).$$
I will give you hint instead of a complete answer since it's a homework question.