Question: 1. (10 points) Prove the following statement. Let C be a countably infinite set of non-negative real valued functions defined on R . Suppose that

1. (10 points)

Prove the following statement. LetC be a countably infinite set of non-negative real valued functions defined on R.

Suppose that for any sequence(gk)k=1 in C, for any sequence(ak)k=1 in R,

sup{k=1ngk(ak))n=1,2,...}<.

Then for any sequence(xk)k=0 in R, limksup{g(xk)gC)}=0.

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