I was reading the proof of a theorem and at a point of the argument the author just posed an inequality which I was unable to derive, but is probably just a smart application of the triangle inequality.
The problem is as follows. Let $f,g:\mathbb{R} \to \mathbb{R}$ be uniformly continuous and $\epsilon, \delta >0$ such that if $x,y\in \mathbb{R}$ are st $|x-y| < \delta$, that then $|f(x)-f(y)|, |g(x)-g(y)| <\epsilon$. Let $I \subset \mathbb{R}$ be an interval with $diam(I) < \delta$
The author then states without proof that $\sup_{x,y\in I} |f(x)-g(y)|^p < |f(t)-g(t)+2\epsilon|^p$ for all $t \in I$., $p\geq 1$
It feels as if this should be easy to prove, but I have been stuck at it for quite some time now. Any help would be greatly appreciated.