Homework 2 Solution

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Problem 6 (Exercise 2.14). Give an example of an open cover of the segment (0, 1) which has no finite subcover.

Problem 7 (Exercise 2.18). Is there a nonempty perfect set in R1 which contains no rational number?

Problem 8 (Exercise 2.20). Are closures and interiors of connected sets always connected?

Problem 9 (Exercise 3.3). If s1 = 2, and

sn+1 = 2 + sn (n = 1, 2, 3, . . . ),

prove that {sn} converges, and that sn < 2 for n = 1, 2, 3, . . . .

Problem 10 (Exercise 3.7). Prove that the convergence of an implies the convergence of

an n

if an 0.

2


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