I am working on a probability problem in which there are $n$ events $E\_{1},E\_{2},...,E\_{n}$. I'm interested to know the value or an upper bound of $P(E\_{1}\cap E\_{2}\cap ... \cap E\_{n})$. I've shown that, for these events $E\_i$, it holds that 

$$P(E\_{i}\cap E\_{j})\leq P(E\_{i})P(E\_{j}), \quad 1\leq i<j\leq n \tag{$1$}$$

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> By using $(1)$, can we conclude $P(E\_{1}\cap E\_{2}\cap ... \cap E\_{n})\leq \Pi\_{i=1}^{n}P(E\_{i})$?
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For example, let
$\Omega = \{1,2,3,4,5,6\}, E\_{1} = \{1,2,3\}$ and $E\_{2}=\{1,4,5,6\}$. Then $E\_{1}\cap E\_{2} = \{1\}$, and $$P(E\_{1}\cap E\_{2})=\frac{1}{6}\leq P(E\_{1})P(E\_{2})=\frac{1}{2}\cdot\frac{2}{3}=\frac{1}{3}$$