TITLE: Borel-Cantelli's II lemma and the independence assumption
QUESTION [5 upvotes]: Let $E_{1},E_{2},....$ be a sequence of independent events. If $\displaystyle\sum_{n=1}^{\infty}\mathcal{P}(E_{n})=\infty$ then $\mathcal{P}(\displaystyle\limsup_{n}E_{n})$=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?

REPLY [5 votes]: Let $A$ some event with $0 < P(A) < 1$ and set $E_n := A$ for $n \in \mathbb N$. Then $(E_n)$ is dependent, and $\sum P(E_n) = \infty$. But as $\limsup E_n = A$, we have $P(\limsup E_n) = P(A) < 1$.

Addendum: For $n \ne m$ we have 
$$ P(E_n \cap E_m) = P(A \cap A) = P(A) \ne P(A)^2 = P(E_n)P(E_m), $$
(as $P(A) \notin \{0,1\}$), so $E_n$ and $E_m$ are dependent. On the other hand, for $n \in \mathbb N$:
\begin{align*}
  \sum_{k \ge 1} P(E_k) &\ge \sum_{k=1}^n P(E_k)\\
          &= n \cdot P(A)
\end{align*}
as hence $\sum_k P(E_k) = \infty$, as $P(A) > 0$.