Let $Z_1, Z_2,\ldots$ be independent identically distributed (i.i.d) binary variables with $P[Z_i = 1] = 1-\alpha $ for some $\alpha > \frac{1}{2}$. Using the transformation $X_i=2Z_i-1$ together with the known facts on the recurrence of random walk on the integers, show that
$$P\left[\frac{1}{n}\sum_{k=1}^n Z_k \le \frac{1}{2}\text{ for infinitely many }n\right]=0$$
I'm thinking I need to use the first lemma of this because of the "infinitely many" term. However, I'm unable to connect this to random walks. Would appreciate all / any input from the community.
Hint: In the random walk, $X_i$ is step #$i$ (to right if $+1$, left if $-1$). Since $\alpha > 1/2$, this is a biased random walk. What does the condition on $\sum_{k=1}^n Z_k$ say about $\sum_{k=1}^n X_k$ (which is where the random walk ends up after $n$ steps)?