→ 3. Let S CR. Suppose {f} and {n} are sequences of functions such that fr 9n →g uniformly on S. f uniformly on S and (a

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3 Let S Cr Suppose F And N Are Sequences Of Functions Such That Fr 9n G Uniformly On S F Uniformly On S And A 1 (33.02 KiB) Viewed 40 times

→ 3. Let S CR. Suppose {f} and {n} are sequences of functions such that fr 9n →g uniformly on S. f uniformly on S and (a) Suppose fn is bounded on S for each n € N. Then show that {f} is uniformly bounded on S, which means that there exists MER such that fn(z)| ≤ M for all r € S and for all n € N. (b) Prove that fn +9n → f+g uniformly on S. (c) Prove that if fr and 9₁ are bounded on S for each n € N, then fn9n → fg uniformly on S.