A uniform distribution problem

Question

Let $(X_{n})$ sequence of random variables such that $X_{n}\sim U[0,a_{n}]$ with $a_{n}>0$. Show that if $a_{n}=n^2$, then a.s only a finite number of the $X_n$ take values $<1$ and if $a_{n}=n$, then a.s an infinite number of $X_{n}$ take values $<1$. I don't quite understand the part "then a.s only a finite number of the $X_n$ take values" and "then a.s an infinite number of $X_{n}$ take values $<1$". Any hint is appreciated.

Answer 1

If $a_n =n^{2}$ then $\sum P(X_n<1)=\sum \frac 1 {n^{2}} <\infty$. Borel-Cantelli Lemma shows that $X_n<1$ infinitely many times with probability $0$.

The second part requires independence of $(X_n)$. With the independence we get the conclusion by the same argument noting that $\sum \frac 1n =\infty$.