Proving a monotonic sequence is unbounded

[alligatorman]

I'm trying to prove that the sequence
$$x_1,x_2,_\cdots$$
of real numbers, where
$$x_1=1$$ and $$x_{n+1}=x_n+\frac{1}{x_n^2}$$ for each $$n=1,2, \cdots$$

is unbounded.

(sorry for the ugly latex! i don't know if there's a way to format that better)

I'm thinking of proving by contradiction, assuming it is bounded and then somehow getting it to imply that the sequence is not increasing, but I'm not sure how to go about it.

Any hints?

 

[morphism]

If it were bounded, it would converge.

 

[HallsofIvy]

And if it were to converge to, say, x, that limit would satisfy
$$x= x+ \frac{1}{x}$$
What values of x satisfy that?

 

[alligatorman]

And if it were to converge to, say, x, that limit would satisfy
$$x= x+ \frac{1}{x}$$
What values of x satisfy that?

I don't understand why this is true.

 

[HallsofIvy]

Start with $x_{n+1}= x_n+ 1/x_n$ and take the limit, as n goes to infinity ,of both sides. If the sequence ${x_n}$ converges to some number, x, then each "$x_n$" or "$x{n+1}$" term will go to x.