
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.

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.