It is well-known that a random walk that starts from $0$ and goes $+1$ or $-1$ at each step with equal probability is expected to end within $\pm O(\sqrt{n})$, where $n$ is the number of steps.
Is there a result that says that with probability approaching $1$ as $n$ grows, the random walk will end within $\pm \sqrt{n}$? For example, does it follow from the central limit theorem?