Homework for Sections 1.4–1.6
Mathematics 360
due Friday, September 27
Read Elementary Classical Analysis, pp. 49–63, and the proofs on pp. 86–93.
1. (The contraction principle)
(a) Suppose that {xn}is a sequence such that, for every n∈N,
|xn+2 −xn+1|< r|xn+1 −xn|
for some r∈]0,1[. Prove that {xn}is a Cauchy sequence. (Hint: Example 1.4.8.)
(b) Give an example of a sequence {xn}satisfying
|xn+2 −xn+1|<|xn+1 −xn|
for every n∈N, but such that {xn}is not a Cauchy sequence.
2. Exercise 25, page 99 of ECA.
3. Given a sequence xn, define the mean sequence by
mn=1
n(x1+x2+· · · +xn)
Show that if xn→x, then mn→x. Find an example where mnconverges but xndoes
not.
4. Exercises 2 and 5, page 56 of ECA. (Prove or find a counterexample.)
5. Construct a sequence with infinitely many distinct cluster points, or prove that this is
impossible.
6. Construct a sequence which does not have a limit, but such that every subsequence
does have a limit; or prove that this is impossible.
7. Exercise 9, page 98 of ECA.
8. Exercise 1, page 63 of ECA.