= Solution
This is the <simple symmetric random walk> on the integers, starting at zero. It is recurrent: it returns to zero infinitely often <almost surely>, and in fact visits every integer infinitely often. To see the two-sided unboundedness explicitly, <gambler's ruin> gives, for positive integers $a,b$,
$$
\mathbb P_0(T_b<T_{-a})=\frac{a}{a+b}.
$$
Letting $a\to\infty$ shows that $b$ is hit with probability one; the analogous argument hits every negative integer. From any state the same argument gives return to zero with probability one. Repeated use of the <Strong Markov property> then gives infinitely many returns and visits. Therefore
$$
\boxed{\limsup_{n\to\infty}X_n=+\infty,\qquad
\liminf_{n\to\infty}X_n=-\infty\quad\text{almost surely}.}
$$
In particular $X_n$ has no limit <almost surely>, and $|X_n|$ does not tend to infinity because of the returns to zero. In contrast, the <strong law of large numbers> gives $X_n/n\to0$ <almost surely>, while the <central limit theorem> gives $X_n/\sqrt n\xrightarrow dN(0,1)$. These distinguish oscillation of the path from scaling limits of its <probability distribution>.
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