Proof of convergence implication

Question

I am trying to show that if $(Z_t)_{t> 0}$ is a stochastic process the following implication holds $$ P(sup_{0<s\le t} |Z_s| > \alpha ) \stackrel{t\to 0}{\to} 0 \Rightarrow Z_t \stackrel{t\to 0}{\to} 0 \ a.s. $$ I am really clueless. Any hint / help is appreciated.

Answer 1

Okay, so we can deduce there exists a deterministic sequence $t_n$ for which the following holds: $$ P(\underbrace{\sup_{0\le s\le t_n}|Z_s|}_{X_n}>\tfrac{1}{n})\le \tfrac{1}{n^2} $$ Now, Borel Cantelli yields that almost surely $X_n\le \tfrac{1}{n}$ for $n$ big enough. That should do that job, I think...