Here is the definition I am given to equicontinuity:
I found this question here:
Is an equicontinuous family of uniformly continuous functions necessarily uniformly equicontinuous?
but I do not understand the definition of an equicontinuous family of continuous functions to be uniformly equicontinuous from it, could anyone tell me the definition of uniformly equicontinuous family of functions (in a way similar to the definition given above (without using small n)), please?
Now, I want to show that: an equicontinuous family of continuous functions on a compact metric space is uniformly equicontinuous.
And I found this question and its solution online:
And I found not less than 3 questions here on this site that answers a question similar to mine but not exactly mine(I have read them all), so could anyone help me proof this question: an equicontinuous family of continuous functions on a compact metric space is uniformly equicontinuous. and tell me how I will use compactness of the given metric space in the proof?
Best Answer
See lemma 1 in my note here, which establishes this. The weaker notion is more pointwise; in its metric formulation:
$$\forall x\forall \varepsilon \exists \delta>0: \forall f \in \mathcal{F}: \forall y: (d(x,y) < \delta) \to \rho(f(x),f(y)) < \delta$$
so that we can find, given $x$,and $\varepsilon$, a $\delta$ that works for all functions at $x$, while the notion you defined is uses the same $\delta$ for all pairs $x,y$ and all $f$ at the same time. It's a similar difference between continuity at $x$ and uniform continuity.
You'll need to know the Lebesgue number lemma for the proof, as a basic fact about compact metric spaces.