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:
(a) Explain why, if the distance from the bottom of the first fold is , the distance from the bottom of the st fold is given by .
(b) Prove that for all , and use this to show that
for all . (Use induction on .)
(c) Deduce that the subsequences and both converge to .
(d) Which standard theorem in these notes allows you to conclude from (c) that as ?
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 , 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... ;)
Let and . By considering the subsequences and show that .