Let $\displaystyle E_{1},E_{2},....$ be a sequence of independent events. If $\displaystyle \displaystyle\sum_{n=1}^{\infty}\mathcal{P}(E_{n}) =\infty$ then $\displaystyle \mathcal{P}(\displaystyle\limsup_{n}E_{n})$=1 i.e the probability that infinitely many events occur is 1.

What easy example one can give to show that the lemma fails if the dependence condition is omitted?

Please step by step, I am beginner in this field...