Why induction cannot be used for infinite sets? Explain why induction cannot be used to conclude: $$\left(\bigcup_{n=1}^\infty A_n\right)^c = \bigcap_{n=1}^\infty A_n^c$$

You should look at how your text defines the set $\left( \bigcup_{n=1}^\infty A_n\right)$.  My guess is that it defines as the set of all elements $x$ for which there exists an $A_n\in \{A_i\}_{i=1}^\infty$ with the property that $x\in A_n$.
To expand on the previous by Pinilla we can't conclude that $\sum_{i=1}^\infty \frac{1}{i} $ is finite even though $\sum_{i=1}^N \frac{1}{i}$ is finite for every positive integer $N$.  
