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hw4solutions

# hw4solutions - Math 318 HW#4 Solutions 1 Abbott Exercise...

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Math 318 HW #4 Solutions 1. Abbott Exercise 7.4.6. Review the discussion immediately preceding Theorem 7.4.4. (a) Produce an example of a sequence f n 0 pointwise on [0 , 1] where lim n →∞ 1 0 f n does not exist. Answer. Consider the function f n whose graph is given in Figure 1 . n 2 1 /n 2 /n Figure 1: The function f n For any x [0 , 1], either x = 0 and f n ( x ) = 0 for all n , or f n ( x ) = 0 for all n > 2 /x ; either way, f n ( x ) 0, so we see that f n 0 pointwise. On the other hand, 1 0 f n = 1 2 2 n · n 2 = n, so lim n →∞ 1 0 f n diverges to + . (b) Produce another example (if necessary) where f n 0 and the sequence 1 0 f n is un- bounded. Answer. The same example from (a) still works. (c) Is it possible to construct each f n to be continuous in the examples of parts (a) and (b)? Answer. The example in (a) is already continuous. (d) Does it seem possible to construct the sequence ( f n ) to be uniformly bounded? Answer. Certainly we can’t get 1 0 f n to be unbounded if the f n are uniformly bounded: if there’s a uniform bound M such that | f n ( x ) | ≤ M for all n and all x [0 , 1], then 1 0 f n M (1 - 0) = M for any n . Also, problem 2 below implies it is not possible to have 1 0 f n diverge at all when the convergence is uniform on any set of the form [ δ, 1] (as it would be for, say, a power series centered at x = 1).

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hw4solutions - Math 318 HW#4 Solutions 1 Abbott Exercise...

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