Proving that a sequence is always positive given two constraining relations

[Mr Davis 97]

Homework Statement

Given that $t_1 = 1$ and $\displaystyle t_{n+1} = \frac{t_n^2 + 2}{2t_n}$ for $n \ge 1$. Show that $t_n > 0$ for all $n$.

Homework Equations

The Attempt at a Solution

Intuitively this is obvious. Since $t_1$ is positive, so is $t_2$, and so on. But I am having trouble proving this rigorously...

 

[fresh_42]

How about an induction?

 

[Mr Davis 97]

How about an induction?

Ah I see, that's a really easy induction. So, suppose I am assuming that $t_n$ converges, and I take the limit of both sides to solve the equation to get that $\lim t_n = \pm \sqrt{2}$. Would I need to prove using induction that $\forall n ~ t_n > 0$ before I can disregard the negative root?

 

[fresh_42]

You said you want a formal proof for $t_n > 0$ which is either an induction or a proof by contradiction. You said nothing about the limit. If all sequence elements are positive, which I think doesn't need a formal proof, the limit can lowest be zero. Otherwise you would have a gap around the limit, i.e. a neighborhood without sequence elements.

 

[Ray Vickson]

Ah I see, that's a really easy induction. So, suppose I am assuming that $t_n$ converges, and I take the limit of both sides to solve the equation to get that $\lim t_n = \pm \sqrt{2}$. Would I need to prove using induction that $\forall n ~ t_n > 0$ before I can disregard the negative root?

If you have a sequence $\{ t_n \}$ with all $t_n > 0$ then if a limit exists it cannot be $< 0$; it may be zero or it may be positive. Think about it, and make sure you understand why.

However, this question (as written down by you) does not ask at all about limits.