Finding a limit of a monotonically decreasing function

Let $a_n$ be a series so $a_1=1$ and $2a_{n+1}<a_n<3a_{n+1}$. Find the limit of $a_n$ when $n\to \infty$.

I have proved that $a_n>0$ and that it monotonically decreasing. But how I find the limit now? Should I use the Squeeze theorem? I know how to solve it when for example we have $a_n = -a_{n+1} + 1$ (we do: $L=\lim_{n\to \infty}a_n $ and $L=-L+1$). But I don't have an equation here.