I don't have my book with me at the moment (and I don't trust myself to do the full proof without it, no matter how elementary), so I won't do the whole thing, but think about this as an induction proof. De Morgan's Law holds for two sets. Can you think of a way to show that if it holds for N sets that it will also hold for N+1? (Given that it holds for 2 sets.)
-Dan
Note: I'm having some thoughts about this for $\displaystyle N \to \infty$. I don't know if you might not run into trouble with that. But as your problem specifies an "n" as an upper index you should be fine for your proof.
There are two basic ways to prove this.
First by DeMorgan quantification rules:
$\displaystyle \begin{array}{rcl}
x \notin \left( {\bigcap\limits_{i = 1}^K {E_i } } \right) & \Leftrightarrow & \left[ {\exists k,1 \le k \le K} \right]\left( {x \notin E_k } \right) \\
& \Leftrightarrow & \left[ {\exists k,1 \le k \le K} \right]\left( {x \in \left( {E_k } \right)'} \right) \\
& \Leftrightarrow & x \in \left( {\bigcup\limits_{i = 1}^K {\left( {E_i } \right)'} } \right) \\
\end{array}.$
The other is by induction.
$\displaystyle \left( {E_1 \cap E_2 } \right)' = \left( {E_1 ' \cup E_2 '} \right).$
Then note $\displaystyle \left( {\bigcap\limits_{i = 1}^{K + 1} {E_i } } \right) = \left( {\bigcap\limits_{i = 1}^K {E_i } } \right) \cap E_{K + 1} .$