Homework 6 Solution

Description

Problem 26 (Exercise 7.2). If {f_n } and {g_n} converge uniformly on a set E, prove that {f_n + g_n} converges uniformly on E. If, in addition, {f_n} and {g_n} are sequences of bounded functions, prove that {f_ng_n} converges uniformly on E.

Problem 27 (Exercise 7.3). Construct sequences {f_n}, {g_n} which converge uniformly on some set E, but such that {f_ng_n} does not converge uniformly on E (of course, {f_ng_n} must converge on E).

Problem 28 (Exercise 7.6). Prove that the series

^∞ ₍₋₁₎n ^{x2 + n}

_n2

n=1

converges uniformly in every bounded interval, but does not converge absolutely for any value of x.

Problem 29 (Exercise 7.11). Suppose {f_n}, {g_n} are defined on E, and

1. f_n has uniformly bounded partial sums;

2. g_n → 0 uniformly on E;

3. g₁(x) ≥ g₂(x) ≥ g₃(x) ≥ . . . for every x ∈ E. Prove that f_ng_n converges uniformly on E.

Problem 30 (Exercise 7.16). Suppose {f_n} is an equicontinuous sequence of functions on a com-pact set K, and {f_n} converges pointwise on K. Prove that {f_n} converges uniformly on K.

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