Homework 8, Math 347, Prof. Eugene Lerman
Due Friday, March 28, 2008
A sequence {an}is monotone if it is either non-increasing or non-decreasing.
Recall that monotone bounded sequences always converge.
1Let {an}and {bn}be two monotone sequences. Is it true that
cn=an+bnis monotone? Prove or give a counterexample.
2Give an example of a sequence of real numbers so that it is
(a) Cauchy but not monotone;
(b) monotone but not Cauchy;
(c) bounded but not Cauchy.
3Let {an}be a convergent sequence of real numbers with an≥0 for all n.
(a) Prove that if limn→∞ an= 0 then limn→∞ √an= 0 as well.
(b) Prove that if limn→∞ an=L > 0 then limn→∞ an=√L.
Hint: for all x, y > 0
|√x−√y|=|√x−√y||√x+√y|
|√x+√y|=|x−y|
√x+√y≤|x−y|
√y
4Let {an}be a sequence defined recursively by a1= 1, an+1 =√an+ 1
for all n≥1.
(a) Prove (by induction): an<2 for all nand an< an+1 for all n. Con-
clude that limn→∞ anexists (What theorem are you using?).
(b) Use the equation an+1 =√an+ 1 to show that L2=L+ 1. What is L?
Hint: Previous problem may be useful for computing limn→∞ √an+ 1.
5Prove that a sequence of complex numbers {zn}converges to L∈Cif
and only if ( Re(zn)→Re(L) and Im(zn)→Im(L)) .
1