Define a stopping time $\tau:= \min\{t:W_{t} = 1 \text{ or } W_{t} = -1 \}$, I want to prove that $\mathbb{P}(\tau < \infty ) = 1$, where $W_{t}$ is a Brwonian motion. Here is my derivation: $$ \begin{aligned} \mathbb{P}(\tau < t ) = \mathbb{P}(\tau < t | W_{t} = 1 \text{ before } W_{t} = -1)P(W_{t} = 1 \text{ before } W_{t} = -1) + \mathbb{P}(\tau < t | W_{t} = -1 \text{ before } W_{t} = 1)P(W_{t} = -1 \text{ before } W_{t} = 1) \end{aligned} $$ Then, $$ \mathbb{P}(\tau < t | W_{t} = 1 \text{ before } W_{t} = -1) = \mathbb{P}(\tau_{1} < t ) \\ \mathbb{P}(\tau < t | W_{t} = -1 \text{ before } W_{t} = 1) = \mathbb{P}(\tau_{-1} < t ) $$ where $\tau_{1}:= \min\{t:W_{t} = 1\}$ and $\tau_{-1}:= \min\{t:W_{t} = -1\}$. Thus, we can only calculate
$$ \begin{aligned} \mathbb{P}(\tau_{1} < t ) &= \mathbb{P}(\tau_{1} < t , W_{t} > 1) + \mathbb{P}(\tau_{1} < t , W_{t} < 1) \\ &= 2 \mathbb{P}(\tau_{1} < t , W_{t} > 1){\text{ (By Reflection principle)}} \\ &=2\mathbb{P}(W_{t} > 1) \end{aligned} $$ For $\mathbb{P}(W_{t} > 1)$, it is very easy to calculate using definition. Similarly to calculate $\mathbb{P}(\tau_{-1} < t ) $.
The last thing is $P(W_{t} = 1 \text{ before } W_{t} = -1) $ which is $\frac{1}{2}$ by the martingale of $W_{t}$ and optional stopping theorem. Finally, let $t$ goes to $\infty$, we can conclue that $\mathbb{P}(\tau < \infty ) = 1$. Does that right?