Contrapositive and Contradiction · Challenge problem

There is an $\varepsilon > 0$ such that for every $N$ there is an $n \ge N$ with $|a_n| \ge \varepsilon$

There is an $\varepsilon > 0$ such that for every $N$ and every $n \ge N$, $|a_n| \ge \varepsilon$

For every $\varepsilon > 0$ and every $N$ there is an $n \ge N$ with $|a_n| \ge \varepsilon$

There are an $\varepsilon > 0$ and an $N$ such that for every $n \ge N$, $|a_n| \ge \varepsilon$