Let $X_1, X_2, \dots $ be a sequence of independent random variables $\mathcal{F}_\infty$ denote the tail $\sigma$-algebra. If $X$ is a random variable which is measurable with respect to $\mathcal{F}_\infty$, then $X$ is almost surely constant, i.e. there exists a constant $C$ such that $P[X=C]=1$.
It looks like we should use the Borel-Cantelli lemma but I don't know how to get at the point where to use it.
If $A \in \mathcal F_\infty$, then there exists a sequence of sets $B_n \in \sigma(X_1,\dots,X_n)$ such that $I_{B_n} \to I_A$. But also, $A \in \sigma(X_n,X_{n+1},\dots)$. Hence for every $n \ge 1$, the sets $A$ and $B_n$ are independent, that is, $\Pr(A \cap B_n) = \Pr(A)\Pr(B_n)$. Now let $n \to \infty$, and we get $\Pr(A) = \Pr(A)^2$. Therefore $\Pr(A) = 0$ or $1$. Hence for every number $c$ we have $\Pr(X<c) = 0$ or $1$. The only way this can happen is if $X$ is some constant almost surely.
