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Prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions from A has a convergent subsequence in the sup-norm.

Let C_0 be the space of functions f:R --> R such that

lim f(x) = 0 as x goes to infinity and negative infinity

C_0 becomes a metric space with sup-norm

||f|| = sup { |f(x)| : x in R }

Prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions from A has a convergent subsequence in the sup-norm.

My current work is to try to reformulate this into a problem where I can use Ascoli-Azrela theorem. One idea was to regard C_0 as the space of functions on the circle as I have seen a similar construction in topology. However, I am unable to fill in the details.

Solution Preview

Here is a standard technique to contruct such a proof.
Let q_1,q_2,q_3,... be some ordering of the rationals, which exists since they are countable.
Let f_1,f_2,f_3,... be some sequence in A.
We will construct a subsequence g_1,g_2,g_3,... as follows:
Let h^1_1, h^1_2,h^1_3,... be a subsequence of f_i such that h_^1_i(q_1) ...

Solution Summary

This solution is comprised of a detailed explanation to prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions from A has a convergent subsequence in the sup-norm.

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