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 c2n1 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
0
