I have hard time proving the following:

> 
> Let $a\_n$ be a sequence such that $a\_n>0$ for all $n$ and:
> 
> 
> $$\lim\_{n \to \infty}a\_na\_{n+1}=A$$
> $$\lim\_{n \to \infty}a\_na\_{n+2}=B$$
> $$\lim\_{n \to \infty}a\_na\_{n+3}=C$$
> 
> 
> I try to prove or disprove that $A=B=C$.
> 
> 
> 

So far I have managed to prove that $A=B$:

$$A^2=\lim\limits\_{n \to \infty} a\_n a\_{n+1}\cdot\lim\limits\_{n \to \infty}a\_{n+2} a\_{n+3} = \lim\limits\_{n \to \infty}\left(a\_n a\_{n+1} a\_{n+2}a\_{n+3}\right)$$
$$B^2= \lim\limits\_{n \to \infty} a\_n a\_{n+2}\cdot\lim\limits\_{n \to \infty}a\_{n+1}a\_{n+3} = \lim\limits\_{n \to \infty}\left(a\_n a\_{n+1} a\_{n+2}a\_{n+3}\right)$$

So we get that $A^2=B^2$ and because $A,B$ are not negative we get that $A=B$.