Let $(X_1,X_2,...)$ be i.i.d random variables, with $P(X_t=1)=P(X_t=-1)=1/2$. Then
$S_t= \frac{1}{t}\sum_{i=1}^{t}X_i $ is a zero mean random walk. Let $\tau$ be the stopping time corresponding to the first time that $S_t$ hits one, \begin{equation} \tau=\inf\{t \geq 1 \mid S_t \geq 1 \} \end{equation} I am trying to show that $P(\tau = \infty)>0$.
For starters, clearly $P(\tau < \infty)>0$ since $P(\tau=1)=1/2$. Also, the probability that $S_t$ exceeds $1$ infinitely often is zero, since $S_t$ converges to $0$. But I would like to show that with positive probability, $S_t$ can converge to zero without ever exceeding $1$.