= Solution
{c}
Intersecting the probability-one hitting events over all positive integers shows that the <simple symmetric random walk> reaches arbitrarily large positive levels almost surely. The hitting times of levels tending to infinity also tend to infinity, since $|S_n|\leq n$. Hence $\mathbb P(B_+)=1$, and in particular $\mathbb P(B_-\cup B_+)=1$.
Alternatively, the union conclusion, the equal probabilities from symmetry, and the <Kolmogorov zero-one law> force both probabilities to equal one. Thus almost every path has arbitrarily large positive and negative excursions. A nearest-neighbor integer path must pass through zero between a positive and a negative excursion. Such crossings occur after every finite time, giving
$$
\boxed{\mathbb P(B_-)=\mathbb P(B_+)=\mathbb P(A)=1.}
$$
This is recurrence of the <simple symmetric random walk>.
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