Finding the limit of the sequence $a_n\cdot a_{n+1}=n,\,n=1,2,3,\cdots.$ Let $(a_n)_{n>=1}$ be a sequence of real numbers defined by the below recurrence relation:

$$a_n\cdot a_{n+1}=n,\quad n=1,2,3,\cdots.$$

Prove that $\lim_{n\to \infty}a_n=+\infty.$
Edit: $a_1>0$

You can prove by induction that
$$
a_{2n+1} = \frac{(2n)!!}{(2n-1)!!}\, a_1,
\qquad
a_{2n+2} = \frac{(2n+1)!!}{(2n)!!}\, \frac{1}{a_1}.
$$
Consider the subsequence $(a_{2n+1})$. You have that
$$
a_{2n+1} = a_1\, \prod_{k=1}^n \left(1 + \frac{1}{2k-1}\right)
\geq a_1\, \sum_{k=1}^n \frac{1}{2k-1}
$$
and the last sum goes to $+\infty$ as $n\to +\infty$.
An analogous computation shows that also $(a_{2n+2})$ diverges to $+\infty$.
