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
2.
c2n  an and c2n1  bn . Prove that cn  converges to A.
Let { f n } be the Fibonacci sequence f n : 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , ...
satisfying f1 = f2 = 1 and fn+1= fn+ fn-1.
Let {rn } be the sequence of ratios of successive Fibonacci numbers. So rn 
f n 1
and
fn
rn : 1 , 2 , 3/2 , 5/3 , 8/5 , 13/8 , 21/13 , ...
a. Is this sequence monotonic?
b. Prove that rn+1 = 1 + 1/rn and hence prove that if this sequence were
convergent then its limit would be ( 5 + 1)/2.
c. Prove that rn+2 = (2rn + 1)/(rn + 1).
Hence show that the subsequence of odd terms and the subsequence of
even terms are monotonic and bounded.
d. Deduce that (rn) is convergent to the Golden Ratio.
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 
5. Let a be a real number in (0, 1). Let an be the real number obtained by deleting the first n
digits of the decimal expansion of a.
For example, if a = 0.1234567... then a1 = 0.234567.., a2 = 0.34567..., etc.
For which a  (0,1) is the sequence an convergent and what is its limit ?