Assume a sequence of random variables $Z_1,Z_2,Z_3,...$ such that $$\sum_{i=1}^{\infty} P(|Z_i-Z|>ε)<\infty$$, for$ε>0$ , then $Z_i \to Z$ almost surely as $n \to \infty$.
So I understand that if I can prove that $Z_i$ converges in probability then it converges almost surely. To prove it converges in probability, I am thinking of sequence of events $A_1\subseteq A_2\subseteq A_3\subseteq\cdots$ and $A_1$ being the $|Z_{\infty}-Z|>ε$. And then prove all $P(A_i)$ has to euqal to $0$?
No, that doesn't work. Convergence in probability does not imply almost sure convergence. The theorem you want to use is the first Borel-Cantelli lemma. Let $A_i$ be any sequence of events in a probability space. Then
$$ \sum_{i=1}^{\infty} P(A_i) < \infty \implies P(A_i\ \text{infinitely often}) = 0 $$
Can you complete the proof with this?
Also if you want an example where convergence in probability does not imply convergence almost surely look here:
Convergence of random variables in probability but not almost surely.