# What is wrong in my "proof" that $\cup_{k=N}^{\infty} E_k - \cap_{k=N}^{\infty} E_k = \emptyset$

I was doing a problem involving limits of sets, and I wanted to figure out what

$$\bigcup_{k=N}^{\infty} E_k - \bigcap_{k=N}^{\infty} E_k$$

is. I got that it is empty, which is obviously false (for instance, $$E_k = \{k \}$$ is a counterexample). The following is my "proof." Could you please take a look and point out the mistake? Thanks.

We begin with the original:

$$\bigcup_{k=N}^{\infty} E_k - \bigcap_{k=N}^{\infty} E_k.$$

Definition of set difference

$$\left(\bigcup_{k=N}^{\infty} E_k \right) \bigcap \left( \bigcap_{k=N}^{\infty} E_k\right)^c$$

DeMorgan's Law

$$\left(\bigcup_{k=N}^{\infty} E_k \right) \bigcap \left(\bigcup_{k=N}^{\infty} E_k^c \right)$$

Distribution Law

$$\bigcup_{k=N}^{\infty} \left[ E_k \bigcap \left(\bigcup_{k=N}^{\infty} E_k^c \right) \right]$$

Distribution Law again

$$\bigcup_{k=N}^{\infty}\left[ \bigcup_{k=N}^{\infty} E_k \cap E_k^c\right]$$

$$=\bigcup_{k=N}^{\infty}\left[ \bigcup_{k=N}^{\infty} \emptyset\right]$$

$$= \emptyset$$

You used the same variable $$k$$ for both the union and the intersection that makes a big difference. Use two different variables and you will see where the mistake is.