3. Let S SR. Suppose {{n} and {9n} are sequences of functions such that fn + f uniformly on S and 9n →g uniformly on S.

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3 Let S Sr Suppose N And 9n Are Sequences Of Functions Such That Fn F Uniformly On S And 9n G Uniformly On S 1 (18 KiB) Viewed 27 times

3. Let S SR. Suppose {{n} and {9n} are sequences of functions such that fn + f uniformly on S and 9n →g uniformly on S. (a) Suppose for is bounded on S for each n E N. Then show that {fr} is uniformly bounded on S, which means that there exists M ER such that fn() < M for all re S and for all neN. (b) Prove that fn +n + f +g uniformly on S. (e) Prove that if fn and 9n are bounded on S for each n e N, then fn9n fg uniformly on S. n