# Using subsequences to prove convergence

Exercise COV.1

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:

• First a fold is made along a horizontal line distance $x$ from the bottom of the paper (where $0 < x < 1 2$ and $x$ is chosen at random). The paper is unfolded and turned upside down.
• Next, the paper is folded so that the bottom edge lies exactly on the last fold just made. The paper is unfolded and turned upside down once again.
• The process in the last paragraph is repeated.

(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 COV.2

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 $a n → 1 + 5 2$.

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