# Prove that the Series of Functions Converges Uniformly

(See attached file for full problem description with equations)

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9.3-5

Let {f_n} (from n - 1 to infinity) be a sequence of functions on [a,b]

such that (f_n)'(x) exists for every x is an element of {a,b](n is an element of I) and

(1) {(f_n)(x_0)} (from n=1 to infinity) converges for some x_0 is an element of [a,b].

(2) {f_n'}(from n=1 to infinity) converges uniformly on [a,b].

Prove that {f_n}(from n = 1 to infinity) converges uniformly on [a,b].Show how this result may be used to weaken that hypothesis of 9.3I.

[Hint: For x is an element of [a,b] write

(f_n)(x) - (f_m)(x) = {[f_n (x) - f_n (x)] - [f_n (x_0) - f_m (x_0)]} + [f_n (x_0) - f_n (x_0)]

Apply 7.7A to obtain

(f_n)(x) - (f_m)(x) = {[f'_n (c) - f'_m (c)](x - x_0) + [f_n (x_0) - f_m (x_0)(x_0)].

Theorem 9.3I

If (for each n is an element of I) f'_n (x) exists for each x is an element of [a,b], if {f}(n=1 to infinity) is continuous on [a,b], if converges on [a,b] to f, and if [f'_n}(from n=1 to infinity) converges uniformly on [a,b] to g, then g(x) = f'(x) (a <= x <= b).

.

That is, lim f'_n (x) = f' (x) (a <= x <= b)

We use the book Methods of Real Analysis by Richard Goldberg.

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#### Solution Summary

This solution is comprised of a detailed explanation to show how this result may be used to weaken that hypothesis of 9.3I.

Differentiability, Bounded Above and Supremums

1. Let A and B be two nonempty sets of real numbers. Define A+B = {a+b: a belongs to A and b belongs to B}. (a) Show that if A is open, then A+B is open. (b) If A and B are both closed, is A+B closed? Justify your answer.

2. Let f be differentiable for x > a and A as x --> infinity. Prove that there is a sequence x_n --> infinity such that f'(x_n) --> A. Give an example to show that f'(x) may not tend to A as x --> infinity.

3. Let f be differentiable for x > a and f(x) + f'(x) --> A as x --> infinity. Prove that f(x) --> A and f'(x) --> 0.

4. If A and B are nonempty subsets of R that are bounded from above, prove that the sup(A+B) = sup(A)+sup(B)

5. Let the points of any countable subset E of (a,b), which may be dense, be arranged in a sequence {x_n}. Let {c_n} be a sequence of positive numbers such that converges. Define f(x) = SUM (a < x <b) (i.e., sum over those indices n for which x_n < x). Verify the following properties of f:

(a) monotonically increasing on (a,b);

(b) discontinuous at every point of E; in fact, .

(c) continuous at every other point of (a,b).

6. Let A = {(x, y) 0 <= x, y <= 1}. If f is a continuous function from A to R, can f be one-to-one? Justify your answer.

7. If SUM a_n converges and if {b_n} is monotone and bounded, prove that SUM a_n b_n converges.

8. Prove that f(x) = x^1/2 is uniformly continuous on [0,infinity).

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