# Uniform continuity: $\delta < \epsilon$

I am reading a proof provided in Rudin's Principles of Mathematical Analysis regarding the Reimann-Stieltjes integral.

In order to prove the statement (non relevant for the question itself) he assumes that a function $$\phi$$ is continuous in a compact interval $$[m,M]$$, hence we know that the function will be uniformly continuous.

Here is where things become strange, (theorem 6.11.)

"Choose an arbitrary $$\epsilon>0$$. Since $$\phi$$ is uniformly continuous on the interval, there exists $$\delta>0$$ such that $$\delta<\epsilon$$ and $$|\phi(s)-\phi(t)|<\epsilon$$ if $$|s-t|\leq \delta$$ and $$t,s \in [m,M]$$"

I didn't know that one requirement for the Uniform continuity was that $$\delta<\epsilon$$. Is there something I'm missing?

We can replace $$\delta$$ by any smaller number in the definition of uniform continuity. It is often useful to make further assumptions like $$\delta <1,\delta <\epsilon$$ etc.