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:

- First a fold is made along a horizontal line distance $x$ from the bottom of the paper (where $0<x<\frac{1}{2}$ 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 ${a}_{n+1}=\frac{1}{2}(1-{a}_{n})$.

(b) Prove that ${a}_{n+2}=\frac{1}{4}(1+{a}_{n})$ for all $n$, and use this to show that

$${a}_{2k+1}=\frac{1}{3}+\frac{x}{{4}^{k}}-\frac{1}{3\xb7{4}^{k}}$$and

$${a}_{2k+2}=\frac{1}{3}+\frac{1}{{4}^{k}}\left(\frac{1-x}{2}\right)-\frac{1}{3\xb7{4}^{k}}$$for all $k\u2a7e1$. (Use induction on $k$.)

(c) Deduce that the subsequences ${b}_{n}={a}_{2n+1}$ and ${c}_{n}={a}_{2n+2}$ both converge to $\frac{1}{3}$.

(d) Which standard theorem in these notes allows you to conclude from (c) that ${a}_{n}\to \frac{1}{3}$ as $n\to \infty $?

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 $\frac{1}{3}$, 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 ${a}_{n+1}=1+\frac{1}{{a}_{n}}$. By considering the subsequences ${a}_{2n}$ and ${a}_{2n+1}$ show that ${a}_{n}\to \frac{1+\sqrt{5}}{2}$.