Prove random variable convergence

Question

I need to prove that $X_n$ converge almost surely, and find it's limit(they are of course independent). I know that density function is $g_n(x)=n x^{n-1}\mathbb{1}_{(0,1]}(x)$. Definition of a.s. convergence is : $P(\lim_{n\rightarrow\infty}X_n=X)=1$. We can calculate CDF explicitly: $P(X_n<t)=t^n$ for $t\in(0,1]$. So if we go with $n\rightarrow\infty$ we see that $P(X_n<t)\rightarrow 0$ for $t\in(0,1)$ and is $1$ for $t=1$.So if I see it correctly if we take the large $n$ only value we may get is $1$ for X? And now how to show that is convege almost surely?

Answer 1

It is easy to check that if $t<1$, $$\sum_n P(X_n<t)<\infty.$$ By Borel-Cantelli, for each $t<1$, the following happens with probability $1$: only finitely many of $[X_n<t]$ occur. Call this event $E_t$. Now consider the countable intersection $$\bigcap_k E_{1 - 1/k}.$$ It also has probability $1$, and conditional on it, $X_n$ converges to $1$.