Math 310, Homework 4 (Ch 7-9)
Collected Wednesday, April 30th
1. Problems II (page 117), problem 12.
2. Use the formal definition of limit of a sequence to prove that the sequence ๐๐=1
๐2
has limit equal to zero (that is, an is what the book calls a โnullโ sequence).
3. Use the (negation of the) formal definition of limit of a sequence to prove that the
sequence ๐๐=๐2 does not have limit equal to 0 (that is, bn is NOT what the book
calls a โnullโ sequence).
4. Problems II (page 118) problem 16 parts ii, iii and v.
(note: you do not need to provide a proof for the surjective/not surjective part of iii.
You are expected to prove the rest; you can use properties of powers, exponential
and logarithmic functions.)
5. Problems II, page 118, Problem 20 (i)
6. Let ๐:๐ โ ๐ be a function. Prove that
a) ๐ is injective โ ๐
is injective
b) ๐ is injective โ ๐
is surjective