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: $a_1>0$

We have:  $a_1a_2 = 1 $ , $a_2a_3 = 2$ ,..., $a_na_{n+1}$ = $n$.
Thus ,  $a_n = \dfrac{a_na_{n-1}}{a_{n-1}}= \dfrac{n-1}{a_{n-1}}= a_{n-2}\dfrac{n-1}{a_{n-1}a_{n-2}}=a_{n-2}\dfrac{n-1}{n-2}= \dfrac{(n-3)(n-1)}{(n-2)a_{n-3}}= a_{n-4}\dfrac{(n-1)(n-3)}{(n-2)(n-4)}= ....=\dfrac{(n-1)(n-3)\cdots 3\cdot 1}{(n-2)(n-4)\cdots 4\cdot 2}a_2= \dfrac{(n-1)(n-3)\cdots 3\cdot 1}{(n-2)(n-4)\cdots 4\cdot 2\cdot a_1} $.
Can you manage to find the limit from this point?
