So i have a sequence defined by $a_1 =1$ and $a_{n+1} = a_n + \frac{1}{a_n}$ and i would like to know $\lim_{n\rightarrow \infty} a_n$. I have said that the sequence $a_n$ is unbounded and thus the limit does not exist. But I don't know how to rigorously prove that the sequence is unbounded and is this enough to say that the limit does not exist?
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Inductively, show first that $a_n>0$ for all $n\in\mathbb N$. This shows that $a_{n+1}-a_n=\frac{1}{a_n}>0$, so $(a_n)$ is increasing. If it has a finite limit $l$, then by letting $n\to\infty$, we obtain that $$l=l+\frac{1}{l}\Rightarrow\frac{1}{l}=0,$$ which is a contradiction. Therefore, $a_n\to\infty$.
detnvvp
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We have $$a^2_{n+1}=a^2_n+2+\frac{1}{a^2_n} > a^2_{n}+2$$
Also, it follows easily by induction that $a_n > 0$ for all $n$.
From above, we have that $a^2_n$ is unbounded, so $a_n$ is unbounded as well.
rkm0959
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The sequence is monotonically increasing, therefore it converges iff it is bounded above.
Note every term is greater than or equal to $1$, and suppose it converges to $r \in \mathbb{R}^+$.
Taking the limit as $n \to \infty$ yields:
$$r = r + \frac{1}{r} \implies r^2 = r^2 + 1 \implies 0 = 1$$
Contradiction. Therefore, the sequence does not converge, hence is not bounded above. QED.
Benjamin Dickman
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