# Prove that an equation has a unique (convergent?) solution

I'm stuck on this problem. Any advice is appreciated:

i. Prove that $\forall n \in \mathbb{N}$ the equation $x^n +x - 1 = 0$ has a unique solution $r_n$.

ii. Find the limit of the sequence $(r_n)_{n\geq1}$.

Here's what I have so far (not much, sadly)

I rewrote the equation in i. to get 1 to the other side. I also believe that the solution should be positive for the few numbers that I have tried. I'm thinking about using Newton's Method but I'm not sure how effective it would be. Any advice?

• (i) isn't even true. For $n = 2$, and it appears for all even $n$, there are two distinct solutions. – Mike Pierce Mar 15 '16 at 2:44
• And indeed, when $n$ is even there are always two solutions. – Robert Israel Mar 15 '16 at 2:48
• Wait, I'm sorry. I missed something. It should say "unique positive." my apologies. – Rethink123 Mar 15 '16 at 2:49

Hint for (2): Is $r_n > 1$ or $< 1$? For fixed $x$ in the appropriate interval, what happens to $x^n + x - 1$ as $n \to \infty$ what does that tell you about how $r_n$ relates to $x$?