Mee Seong Im Introduction to Logic and Proofs Monday July 28, 2008
Problem 14.3. [page 287]. For each condition below, give examples of se-
quences <a>and <b>such that lim an= 0, lim bndoes not exist, and the
specified condition holds.
a) lim(anbn) = 0.
b) lim(anbn) = 1.
c) lim(anbn) does not exist.
a) Take an=1
nand bn= (โ1)n.
b) Take an= (โ1)n1
nand bn= (โ1)nn.
c) Take an=1
nand bn= (โ1)nn2. Then lim(anbn) = lim(โ1)nndoes not
exist.
โโโโโโโโโโโโโโโโโโโโโโ
Problem 14.5. Find a counterexample to the following false statement.
โIf an< bnfor all nand Pbnconverges, then Panconverges.โ
This is false. Here is the counterexample: for all ntake an=โ1 and
bn= 0. Then an< bnfor every nand Pbn=P0=0<โ. However,
Pan=Pโ1 = โโ.So Pandiverges.
QUESTION: What would you add to make the above statement true?
โโโโโโโโโโโโโโโโโโโโโโ
For this exercise, determine whether the statement is true or false. If true,
provide a proof; if false, provide a counterexample.
Problem 14.8. Let < x > be a sequence of real numbers.
a) If < x > is unbounded, then < x > has no limit.
b) If < x > is not monotone, then < x > has no limit.
a) True. Pro of for a Contradiction.Suppose < x > has a limit. Then
limnโโ xn=L < โ. Then given any ๎ > 0, there exists N=N(๎)>0
such that for all nโฅN|xnโL|< ๎. This implies that < x > is bounded.
Contradiction. ๎
b) False. Let xn= (โ1)n1
n. Then < xn>is not monotone but lim xn= 0.
โโโโโโโโโโโโโโโโโโโโโโ
Problem 14.15. Suppose that bโคL+๎for all ๎ > 0.Prove that bโคL.
1