chapter7, 25(1)

# chapter7, 25(1) - δ := ±/M we get that | t-s | < δ...

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Special Problem #3 Math 5616 Joshua Miller April 13, 2004 Special Problem #3 :Continuation of 7.26, Verify the following assertions: (b) { f n } is equicontinuous on [0 , 1], since | f 0 n | ≤ M . (c) Some { f n k } converges to some f , uniformly on [0 , 1]. (b) Claim: | f n ( t ) - f n ( s ) t - s | ≤ M for all t, s . If this didn’t hold we could use the Mean Value Theorem to ﬁnd a ξ , say between s and t so that | f 0 n ( ξ ) | > M which would contradict the facts established in part (a). Now our claim holds which means that for any ± > 0 we see that letting
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Unformatted text preview: δ := ±/M we get that | t-s | < δ implies | f n ( t )-f n ( s ) | < | t-s | M < δM < ( ±/M ) M = ± . (c) All the assumptions of Theorem 7.25 in Rudin are satisﬁed. The sequence of f n ’s are deﬁned on the compact set [0 , 1], they are uniformly bounded from part (a) and thus pointwise bounded, and they are equiconintuous from part (b); thus, some { f n k } converges to some f , uniformly on [0 , 1]. ± 1...
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## This note was uploaded on 12/20/2011 for the course MATHEMATIC 201 taught by Professor Joshuamiller during the Spring '11 term at Fudan University.

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