A partial converse of Borel-Cantelli lemma I'm trying to solve this problem in Prof Tao's note but no progress so far:
Let $(E_n)_n$ be a sequence of events with $\inf\limits_n \mathbb P(E_n) > 0$. Show that: $\mathbb P(\sum\limits_{n \geq 0}1_{E_n} = \infty) > 0.$ using $2$ approaches:


*

*If $\mathbb P(E_n) \geq \epsilon > 0, \forall n$, show that: $\mathbb P(\sum\limits_{n \leq N} 1_{E_n} \geq \frac{\epsilon N}{2}) \geq \frac{\epsilon}{2}$.

*Applying Fatou's lemma to the random variable $1_{\limsup E_n}$.
My work so far for the second approach:
$\inf\limits_n \mathbb P(E_n) > 0 \Rightarrow \liminf \mathbb P(E_n) > 0$. Then I don't know how to link this with $\mathbb P(\sum\limits_{n \geq 0}1_{E_n} = \infty)$
 A: As @Did pointed out, there was an error in my answer in the first part. The following is my new attempt (though not complete).
We have $\sum\limits_{n \leq N} 1_{E_n} \geq \frac{\epsilon N}{2} $ iff there are at least $K$ (and at most $N$) numbers $1$, where $K = 1+$ integer part of $\frac{\epsilon N}{2}$ (if $\frac{\epsilon N}{2}$ is integer then there will be at least $K+1$).
Thus, 
$$\mathbb P(\sum\limits_{n \leq N} 1_{E_n} \geq \frac{\epsilon N}{2})  = \sum_{j=K}^N \mathbb P(\sum\limits_{n \leq N} 1_{E_n} =j) \geq \sum_{j=K}^N \mathbb P(\sum\limits_{i = 1, n_i \leq N}^{j} 1_{E_{n_i}} = j) = \sum_{j=K}^N\mathbb P(\cap_{i = 1}^j\{1_{E_{n_i}}=1 \}) $$
$$= \sum_{j=K}^N\prod\limits_{i=1}^j \mathbb P(\{1_{E_{n_i}}=1 \}) = \sum_{j=K}^N\prod\limits_{i=1}^j \mathbb P(E_n) \geq \sum_{j=K}^N\epsilon^j = \frac{\epsilon^K-\epsilon^{N+1}}{1-\epsilon} \geq \frac{\epsilon}{1-\epsilon}(\epsilon^{\epsilon N /2}-\epsilon^N)$$
We need to show the last expression is bigger than $\epsilon/2$. I will complete it later.
For the second part:
$$\mathbb P(\sum 1_{E_n} = \infty) \geq \mathbb P(\{1_{\limsup E_n}=1\})$$
and
$$\mathbb P(\{1_{\limsup E_n}=1\}) = \mathbb P(\{\limsup 1_{E_n}=1\}) = \mathbb P(\limsup \{1_{E_n}=1\})$$
$$ = 1- \mathbb P(\liminf \{1_{E_n}=0\} \geq 1 - \liminf \mathbb P(\{1_{E_n}=0\}) = \limsup \mathbb P (\{1_{E_n}=1\})$$
$$= \limsup \mathbb P(E_n)  \geq \mathbb \inf P(E_n)> 0$$
