Is a sequence diverging almost surely to infinity almost surely positive?

Question

I have proved that a sequence of random variables $(M_n)_{n\in\mathbb N}$ diverges to $+\infty$ almost surely. I.e I have proved that $$\bigcap_{c\in\mathbb Q^+}\bigcup_{N=1}^{\infty}\bigcap_{n=N}^\infty\{M_n>c\}$$ has probability 1 thanks to Borel-Cantelli lemma.

Is that enough to imply that those variables are positive almost surely ?

Thank you!

EDIT : My question actually is, "Is that enough to imply that those variables are positive almost surely for sufficiently large $n$ ?"