Using subsequences to prove convergence - exercises

Exercise.

A student is applying for a job and folds the covering letter (which is written on a sheet of paper of length 1) into thirds to put it in the envelope, using the following method:

image of fold1 image of fold2

(a) Explain why, if the distance from the bottom of the first fold is a 1 = x , the distance from the bottom of the n + 1 st fold is given by a n + 1 = 1 2 ( 1 - a n ) .

(b) Prove that a n + 2 = 1 4 ( 1 + a n ) for all n , and use this to show that

a 2 k + 1 = 1 3 + x 4 k - 1 3 · 4 k

and

a 2 k + 2 = 1 3 + 1 4 k 1 - x 2 - 1 3 · 4 k

for all k 1 . (Use induction on k .)

(c) Deduce that the subsequences b n = a 2 n + 1 and c n = a 2 n + 2 both converge to 1 3 .

(d) Which standard theorem in these notes allows you to conclude from (c) that a n 1 3 as n ?

If you are still struggling with the idea of the definition of convergence think of it as saying whatever the size of the envelope, as long as it is a small amount more than 1 3 , then some number of steps of this process is guaranteed to get the paper folded in a way to get it in the envelope. Of course, your job prospects may also depend on other things too, such as a nice crisp and uncreased covering letter... ;)

Exercise.

Let a 0 = 1 and a n + 1 = 1 + 1 a n . By considering the subsequences a 2 n and a 2 n + 1 show that a n 1 + 5 2 .