Proving a sequence is bounded below.

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence which is eventually positive. Prove that it is bounded below.

For now, my only idea was to say that, "eventually positive" means that there exist $M \in \mathbb{N}$ s.t for all $n>M$ we have $a_{n+M}\ge 0$

From there I think I should work out an $L$, so that I could say that this $L$ exist such that for all $n$, $a_n > L$ but I can't see how I am supposed to link those two.

Eventually positive means that there is $N$ such that for $n>N,\ a_n>0$.
Therefore for all $n$, $a_n\geq \min(a_1,a_2,...,a_N, 0)$.