$\{X_{n}\}$ be a sequence of independent random variables defined on a
probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Prove that the
probability sequence $\{X_{n}\}$ converges is either one or zero.
I am studying for an exam that I have and this is one problem that I am stuck on. I tried to argue by contradiction (i.e. suppose $X_{n} \rightarrow c $ for some $0 < c < 1$), and I tried to use different inequalities but I haven't had any luck. I'm also familiar with Kolmogorov's Zero-One Law, and Borel-Cantelli Lemmas, but I'm not sure how to apply any of them here.
I will appreciate your help in this problem
Best Answer
$(X_n)$ converges iff $(X_k,X_{k+1},...)$ converges, Hence the event $\{\lim_n X_n \text {exists}\}$ is measurable w.r.t. sigma field generated by $X_k,X_{k+1},...$. This is true for each $k$ and hence the event is measurable w.r.t. the tail sigma field. By $0-1$ Law the event has probabilty $0$ or $1$.