1. ## Proof of sequence

I was going over my practice exam for tonights final, and came across the following problem:

Show that the sequence defined by
a1 = 1
and
an+1 = 3 - 1/an

is increasing and an < 3 for all n. Deduce that {an} is convergent and find its limit.

Any time i try to prove the statement i end up assuming that the sequence is increasing in the actual proof.

2. Do you mean: $\displaystyle a_{1} = 1 \qquad a_{{\color{red}n+1}} = 3 - \frac{1}{a_{n}}$

Remember that: Every monotonic, bounded sequence is convergent.

Use induction to show that $\displaystyle \forall n \geq 2 \ : \ 2 \leq a_{n} \leq 3$ (i.e. it's bounded).

And you can use induction again to show that $\displaystyle a_{n+1} > a_{n}$ is true for all n (i.e. it's monotonic).

Now this proves convergence. So, let $\displaystyle \lim_{n \to \infty} a_{n} = L$ (we can assume this because we know $\displaystyle a_{n}$ converges to some limit which we call L).

Now take the limit from both sides of the recursive sequence:

\displaystyle \begin{aligned}\lim_{n \to \infty} a_{n+1} & = \lim_{n \to \infty} \left(3 - \frac{1}{a_{n}}\right) \\ & \ \ \vdots \\ L & = 3 - \frac{1}{L} \end{aligned}

and solve the quadratic, picking the appropriate solution.