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.


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