real analysis - Let $a_n$ be a sequence s.t $a_1 > 0 land a_{n+1} = a_n + dfrac{1}{a_n}$. Prove that $a_n$ is increasing and tends to infinity

Let $a_n$ be a sequence s.t $a_1 > 0 \land a_{n+1} = a_n + \frac{1}{a_n}$. Prove that $a_n$ is increasing and tends to infinity.

Proof:

Consider $a_{n+1} - a_n$:

$a_{n+1} - a_n = a_n + \frac{1}{a_n} - a_n = \frac{1}{a_n}$ This is greater than $0$. Thus, $a_n$ is increasing.

Now this is where I need some help. I would like to say that $a_n$ is unbounded and then conclude that monotone and unbounded implies tending to infinity.

Maybe by contradiction?

Answer

You proved that $a_n$ is increasing. Assume that it is bounded. Then it would follow that $a_n$ is convergent to a real number $L>0$. But taking $n \to \infty$ into the recurrence relation gives
$$ L+\frac{1}{L}=L$$

which is a contradiction. Therefore $a_n$ is unbounded and it follows that $a_n \to \infty$.