Stopping times – Limit of the stochastic process.

Consider any stochastic process. $ {X (t) } ^ {+ infty} _ {t = 1} $ with $ t = 1,2, … $ that diverges to $ + infty $ almost sure
For any $ L in mathbb {R} $, leave $ p (L) $ denotes the probability that $ X (t) <L $ for some $ t $.
Try out $ lim_ {L to – infty} p (L) = 0 $.

I have been trying to prove this for a while for my financial research and I have not managed to do so. Could someone give me an idea? Thank you.