How to compare a sum of uniform RVs with a uniform RV? Let $(X_n)_{n\geq1}$ be a sequence of i.i.d. $\sim \text{Uni}([0,1])$ distributed random variables. I want to show that 
$$\mathbb{P}\big(n^2 X_{n+1} < \sum\limits_{k=1}^n X_k ~\text{for infinitely many } n \in \mathbb{N} \big)=1 $$
This cries out for Borel Cantelli, but I don't know how to use this theorem when there is more than one random variable involved. 
By what means can I compare the two sides?

This is an application of Lévy's conditional form of Borel-Cantelli lemma. The result is often stated as follows. Consider a  sequence of events $(A_n)_n$ which is adapted to a given filtration $(\mathcal F_n)_n$. Then the random series $\sum\limits_n\mathbf 1_{A_n}$ converges/diverges almost surely if and only if the random series $\sum\limits_n\mathrm P(A_{n+1}\mid\mathcal F_{n})$ converges/diverges almost surely.
Here, consider $\mathcal F_n=\sigma(X_k;k\leqslant n)$ and $A_{n+1}=[n^2X_{n+1}\leqslant S_n]$ with $S_n=X_1+\cdots+X_n$. Then $\mathrm P(A_{n+1}\mid\mathcal F_{n})=\frac1{n^2}S_n$. By the strong law of large numbers, $\frac1nS_n\to\mathrm E(X_1)=\frac12$ hence, almost surely, $S_n\gt\frac14n$ for every $n$ large enough. This proves that $\sum\limits_n\frac1{n^2}S_n$ diverges almost surely. Hence, almost surely, infinitely many events $A_n$ occur, QED.
