Tail algebra and tail events in a sequence of i.i.d. random variables 
Suppose $\{X_1,X_2,\dots\}$ is a sequence of i.i.d. random variables and
$P[X_i=1]=P[X_i=-1]=\frac{1}{2}$
Define $S_n=\sum_{i=1}^n X_i$. Let $\bigcap_{n=1}^\infty \sigma\{X_k:k\geq n\} $ be the tail $\sigma-$algebra of $\{X_k\: k \geq 1\}$.
Show that $B_{-}=\{\liminf_{n\to \infty} S_n=-\infty\}$ and $B_{+}=\{\limsup_{n\to \infty} S_n = \infty\}$ are tail events. Moreover $P(B_{-})=P(B_{+})$.

Does $B_{-}$ equal $\{X_i=-1 \text{ for infinitely many }i\}$? And $B_{+}=\{X_i=1 \text{ for infinitely many } i \}$?
How to show $P(B_{-})=P(B_{+})$?
 A: Did points out that the answer to the first question is not true. For example, the sequence $(+1, -1, +1, 1, \ldots)$ is a counter example.
However, seeing that these events are in the tail algbera is still okay -- if $a_i$ is a sequence with $lim sup_{n \to \infty} \Sigma^n a_i = \infty$, then changing the values of the first few $a_i$ only shifts the value of the lim sup by a finite number, so it is still infinity. That is, we only need to know the values of $a_N, a_{N+1}, \ldots$ to check the claim about the lim sup being infinite, and this means that the event is measurable in all $\sigma(X_{N}, X_{N+1}, \ldots)$.
As for why they are equal - reflect across $0$ and work with the iid family $-X_i$. This interchanges $B_{-}$ and $B_{+}$ and leaves the joint distributions the same, so the probabilities of those two events have to be equal. (If you like, work with $\Omega = \{-1,1\}^{\mathbb{N}}$.) (I still think this reasoning is correct.)
What is the probability? The previous reasoning (in the comments), which depended on the false assertion about the number of ones and minus ones, is false. I think maybe instead one could use the central limit theorem to prove that the probability of $P(B_{-})$ is positive. On the other hand, maybe could one could also rescue an argument by contradiction -- if the random walk is almost surely bounded above and below by a (random) $N$ and $-N$, something funny happens?
