16.- Consider the sequence of functions (fr), given by fx: [0,00) — R, fr(t) = sint + 472k2 Prove the following statemen

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16 Consider The Sequence Of Functions Fr Given By Fx 0 00 R Fr T Sint 472k2 Prove The Following Statemen 1 (30.45 KiB) Viewed 16 times

16.- Consider the sequence of functions (fr), given by fx: [0,00) — R, fr(t) = sint + 472k2 Prove the following statements: a) The set H := {fk: K EN } is equicontinuous. b) The set H(t) is relatively compact in IR for each t € [0,-). Hint: Prove that the sequence (fr) converges pointwise to 0 on [0,c.). c) H is not a compact subset of c;([0,-), R). Hint: Prove that the sequence (fr) does not converge uniformly on [0,00). Conclude that the compactness of K is necessary in the Arzelá-Ascoli theorem.