### Uniform continuity and boundedness

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##### Reference no: EM13124343

1. Let f:R->R be uniformly continuous on R (the reals) and let f_n(x)=f(x+1/n) for x in the reals. Show that (f_n) converges uniformly on the reals to f.

2. Suppose that the sequence (f_n) converges uniformly to f on the set A, and suppose that each f_n is bounded on A. (That is, for each n there is a constant M_n such that |f_n(x)| < M_n for all x in A). Show that the function f is bounded on A.

3. Let f_n(x) = nx/(1+nx^2) for x in A =[0,infinity). Show that each f_n is bounded on A, but the point-wise limit f of the sequence is not bounded on A. Does (f_n) converge uniformly to f on A? Hint: f_n(1/squareroot(n))=squareroot(n)/2, where squareroot n= n^0.5.

4. Let f_n(x) =(x^n)/n for x in [0,1]. Show that for the sequence (f_n) of differentiable functions converges uniformly to a differentiable function f on [0,1], and that the sequence (f'_n) converges on [0,1] to a function g, but that g(1) does not equal f'(1).

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