Your absolutely right, if the same

were needed then then inf of the deltas could be zero.

But the thing is I don't see why the same delta is needed. i have the following proof ( which I'll summarize in some parts to avoid a long text):

The proof follows much on the lines of the proof given in Elements of Real Analysis, Bartle

Given

( the set of continuous function from K to

where K is compact and a subset of

we want to show that if

is a subset of

)and

is bounded and equicontinuous then every subsequence of a sequence of functions in

.

We can find a set

which is numerable and dense in K, then we can find a sequence of functions which i'll lable

which for every point in A the sequence

converges in

.

The thing is that then from equicontinuity we can find a

which makes every function

, if

and then take open balls with center in each

and radius

the union of these balls complete covers K, and because K is compact we can find an open subcover such that it covers K,using the fact that these functions are Cauchy convergent in

. we find that for [tex]x \in K[\math],

Why can't we take open balls with different radius delta (i.e., use uniform continuity)?