True of false: $|f(x)-g(x)|<\epsilon$ $\forall x\in I$ $\Rightarrow$ $|\sup(f(x))-\sup(g(x))|\leq \epsilon$
I actually have an idea for a proof in case it is correct, but just to make sure what I'll write isn't rubbish I would like to get a confirmation from you all.
Intuitively, it seems logical to me.
If anyone has a counter example I would love to see it.
Thanks a lot.