how to prove a sequence is unbounded?

Question

I'm little bit confused about how to prove that sequences are unbounded. For example, I have this sequence: $$A_1 = a,\quad (a>0, a \in \mathbb{R}),\\ A_{n+1} = A_n + (A_n)^2/(1+A_n)^2$$

I know how to prove that it's an increasing sequence, but from here I get stuck.

i need to show that lim An = infinity

thanx!

Answer 1

The function $f(x) = \dfrac{x^2}{(1+x)^2}$ is increasing on $(0,\infty)$. Since $\{A_n\}$ is increasing you get $$ A_{n+1} - A_n = \frac{A_n^2}{(1 + A_n)^2} \ge \frac{A_1^2}{(1 + A_1)^2} \ge \frac{a^2}{(1 + a)^2}.$$

Answer 2

It is increasing, hence all terms are $\ge a$. The function $f:x\to x+x^2/(1+x^2)$ is continuous on $(a,\infty)$ and has no fixed points.

Assume that the sequence is bounded. Then it is convergent. The limit is a fixed point of $f$. You get a contradiction.

Answer 3

There is a theorem which you must have learnt about increasing bounded sequence. If you use that theorem you should be able to get to contradiction.