Let $(X_n:n=1,2,\ldots)$ be an independent sequence of random variables, where, for each $n$, $X_n$ is uniformly distributed on $[0,n]$. Calculate $P(\{\omega:X_n(\omega)\to \infty \text{ as } n\to\infty\})$.
I just don't know how to solve this problem or what definitions I should use in order to solve it.
Notice that your even is a zero-one event. Define the event $A_n:=\{X_n<c\}$ for some $c>0$. Then
$$\sum_{n\geq 0} P(A_n)=\sum_{n\geq 0} c/n=\infty.$$
So by the reverse Borel Cantelli lemma, $A_n$ occurs infinitely often with probability 1. This implies that $P(\omega: X_n(\omega)\rightarrow\infty)<1$, and since it's a zero-one event, its probability must be 0.