Homework #11
Subsequences
Read section 2.6
Exercises from the text: Page 110: #1, #2, #4, #7,
Other exercises in thought and logic that you will most certainly find interesting.
1. Suppose that an  converges to A and that bn  converges to A. Define the sequence
cn  by that c2n  an and c2n1  bn . Prove that cn  converges to A.
if n  1

 a
2. Let a  0 be a real number and define an  
. For what values of a will
a

a
if
n

1

n

1

this sequence converge? Find the limit for these values of a. Of course you will prove
your conjecture!
3. Show that every sequence has a monotone subsequence.
4. Let an  be any sequence and suppose that S  lim sup an . Prove that there exists a
subsequence of an  converging to S.
n 