= Solution
Represent the <simple random walk> by $X_n=\xi_1+\cdots+\xi_n$, where the increments are <independent and identically distributed> and satisfy $\mathbb P(\xi_k=1)=p$, $\mathbb P(\xi_k=-1)=1-p$. They are integrable and have <expectation> $2p-1$, so the <strong law of large numbers> gives
$$
\boxed{\frac{X_n}{n}\longrightarrow\mu=2p-1\quad\text{almost surely}.}
$$
If $p\ne1/2$, this nonzero drift implies that $X_n$ eventually leaves every fixed finite interval and tends to the appropriate infinity. Hence it visits zero only finitely many times <almost surely>, so zero is a <transient state>. By translation of the starting state, every state is transient: \b[the chain is transient for $p\ne1/2$]. This also covers the deterministic endpoint cases $p=0,1$, without assuming <Markov chain irreducibility> at those endpoints.
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