4130sols8 - 4130 HOMEWORK 8 Due Tuesday May 3 (1) Let fn :...

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4130 HOMEWORK 8 Due Tuesday May 3 (1) Let f n : A R be functions which converge uniformly on A to a function f . Let x 0 be a cluster point of A . Suppose lim x x 0 f n ( x ) exists for all n . Let L n = lim x x 0 f n ( x ). (a) Show that the sequence { L n } converges. We show that { L n } is a Cauchy sequence. Let ε > 0. Since the sequence { f n } converges uniformly, it is uniformly Cauchy, and so there exists N N such that if m,n > N then | f n ( x ) - f m ( x ) | < ε/ 3 for all x A . Let m,n > N . There exists δ 1 > 0 such that if | x - x 0 | < δ 1 , we have | f n ( x ) - L n | < ε/ 3. There exists δ 2 > 0 such that if | x - x 0 | < δ 2 , we have | f m ( x ) - L m | < ε/ 3. By the triangle inequality: | L n - L m | ≤ | L n - f n ( x ) | + | f n ( x ) - f m ( x ) | + | f m ( x ) - L m | for all x A . In particular, we can choose some x with | x - x 0 | < δ 1 2 . Then we get | L n - L m | < ε. Therefore, for all
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4130sols8 - 4130 HOMEWORK 8 Due Tuesday May 3 (1) Let fn :...

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