Homework due Jan. 23:

From Rudin Ch. 7: Exercises 3, 6, 11, 12. Additional exercises:

- (Exercise 9 from Rudin rephrased) Let be a sequence of continuous functions on a set .
- Prove that if converge uniformly to then
for every sequence of points such that and .

- Is the converse true? That is, if () holds whenever does it follow that converge to uniformly?

- Prove that if converge uniformly to then
- Use Theorem 7.11 from Rudin to prove the following result stated in class on Jan. 11:

Let be a doubly indexed sequence such that

uniformly in , i.e., for every there is an so that for we have for all . If for each then converges and

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