# Recursive sequence problem: proofs by mathematical induction

1. Sep 24, 2004

Guys,

I'm trying to prove by induction that the sequence given by
$$a_{n+1}=3-\frac{1}{a_n} \qquad a_1=1$$ is increasing and $$a_n < 3 \qquad \forall n .$$

Is the following correct? Thank you.

$$n = 1 \Longrightarrow a_2=2>a_1$$ is true.

We assume $$n = k$$ is true. Then,

$$3-\frac{1}{a_{k+1}} > 3-\frac{1}{a_k}$$

$$a_{k+2} > a_{k+1}$$ is true for $$n=k+1$$.

This shows, by mathematical induction, that

$$a_{n+1} > a_{n} \qquad \forall n .$$

$$a_1 < 3$$ is true.

We assume $$n=k$$ is true. Then,

$$a_k < 3$$

$$\frac{1}{a_k} > \frac{1}{3}$$

$$-\frac{1}{a_k} < -\frac{1}{3}$$

$$3-\frac{1}{a_k} < 3-\frac{1}{3}$$

$$a_{k+1} < \frac{8}{3} < 3$$

$$a_{k+1} < 3$$ is true for $$n = k+1$$. Thus,

$$a_{n} < 3 \qquad \forall n .$$