Application of Borel Cantelli Lemma in Homework Problem I have a homework problem which I encounter some difficulty in. I sincerely hope you all can advise me in proceeding the steps. 
I am supposed to construct a sequence of real valued independent random variables $X_1, X_2,..$ such that $\sum_{n=1}^{\infty} X_n < \infty$ but $\sum_{n=1}^{\infty} E(X_n)=\infty$, where $E(X_n)$ is the expectation of $X_n$.
I construct the random variables as follows:
$$P(X_n=n)={1\over{n^2}}$$
$$P(X_n=0)=1-{1\over{n^2}}$$ 
In this way, the summation of the expectations will be the standard divergent harmonic series $$\sum_{n=1}^{\infty} {1\over{n}}$$
Using Borel Cantelli Lemma, $\sum_{n=1}^{\infty} P(X_n=n)=\sum_{n=1}^{\infty}{1 \over n^2}< \infty$, which means $P(\left\{X_n=n\right\}$ occurs infinitely many times$)=0$.
On the other hand, $\sum_{n=1}^{\infty} P(X_n=0)=\sum_{n=1}^{\infty}{1-{1 \over n^2}}= \infty$, which means $P(\left\{X_n=0\right\}$ occurs infinitely many times$)=1$.
Now I know that almost surely, $X_n=0$ infinitely many times. However, how do I make it that the tail sum of $\sum_{n=1}^{\infty} X_n$ equal to $0$ so that $\sum_{n=1}^{\infty} X_n<\infty$?
Thank you in advance for your suggestions.   

With probability $1/n^2, X_n=n$. The Borel-Cantelli lemma tells you that this can only happen a finite amount of time. It could be $5,100,34934$ or $2904904838309$ times, the point is there exist a (finite) value $N$ for which $X_n=0$ for all $n> N$. You do not move the $X_n$ that are null to the end, they get there naturally. You can then write
$$
\sum_{n=1}^\infty X_n=\sum_{n=1}^NX_n+\sum_{n=N+1}^\infty X_n.
$$
The first sum on the RHS is finite, it is at most $\sum_{n=1}^N n=N(N+1)/2$, and the second sum is $0$, thus your sum is bounded by $N(N+1)/2<\infty$ for some $N\in \mathbb{N}$.
