ch9 - Sequences of Functions Uniform convergence 9.1 Assume...

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Unformatted text preview: Sequences of Functions Uniform convergence 9.1 Assume that f n f uniformly on S and that each f n is bounded on S. Prove that { f n } is uniformly bounded on S. Proof : Since f n f uniformly on S, then given = 1 , there exists a positive integer n such that as n n , we have | f n ( x )- f ( x ) | 1 for all x S. (*) Hence, f ( x ) is bounded on S by the following | f ( x ) | | f n ( x ) | + 1 M ( n ) + 1 for all x S. (**) where | f n ( x ) | M ( n ) for all x S. Let | f 1 ( x ) | M (1) ,..., | f n- 1 ( x ) | M ( n- 1) for all x S, then by (*) and (**), | f n ( x ) | 1 + | f ( x ) | M ( n ) + 2 for all n n . So, | f n ( x ) | M for all x S and for all n where M = max ( M (1) ,...,M ( n- 1) ,M ( n ) + 2) . Remark : (1) In the proof, we also shows that the limit function f is bounded on S. (2) There is another proof. We give it as a reference. Proof : Since Since f n f uniformly on S, then given = 1 , there exists a positive integer n such that as n n , we have | f n ( x )- f n + k ( x ) | 1 for all x S and k = 1 , 2 ,... So, for all x S, and k = 1 , 2 ,... | f n + k ( x ) | 1 + | f n ( x ) | M ( n ) + 1 (*) where | f n ( x ) | M ( n ) for all x S. Let | f 1 ( x ) | M (1) ,..., | f n- 1 ( x ) | M ( n- 1) for all x S, then by (*), | f n ( x ) | M for all x S and for all n 1 where M = max ( M (1) ,...,M ( n- 1) ,M ( n ) + 1) . 9.2 Define two sequences { f n } and { g n } as follows: f n ( x ) = x 1 + 1 n if x R, n = 1 , 2 ,..., g n ( x ) = 1 n if x = 0 or if x is irrational, b + 1 n if x is rational, say x = a b , b > . Let h n ( x ) = f n ( x ) g n ( x ) . (a) Prove that both { f n } and { g n } converges uniformly on every bounded interval. Proof : Note that it is clear that lim n f n ( x ) = f ( x ) = x, for all x R and lim n g n ( x ) = g ( x ) = 0 if x = 0 or if x is irrational, b if x is ratonal, say x = a b ,b > . In addition, in order to show that { f n } and { g n } converges uniformly on every bounded interval, it suffices to consider the case of any compact interval [- M,M ] , M > . Given > , there exists a positive integer N such that as n N, we have M n < and 1 n < . Hence, for this , we have as n N | f n ( x )- f ( x ) | = x n M n < for all x [- M,M ] and | g n ( x )- g ( x ) | 1 n < for all x [- M,M ] . That is, we have proved that { f n } and { g n } converges uniformly on every bounded interval. Remark : In the proof, we use the easy result directly from definition of uniform convergence as follows. If f n f uniformly on S, then f n f uniformly on T for every subset T of S....
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ch9 - Sequences of Functions Uniform convergence 9.1 Assume...

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