What is the difference between two statements of $\varepsilon-N$ definition? Here is a homework question, TRUE/FALSE:
$$\lim_{n\to\infty}a_n=a\Longleftrightarrow$$


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*$\forall\varepsilon>0,\ \exists N\in\mathbb{Z^+},\ \text{whenever}\ n>N\Rightarrow|a_n-a|<\varepsilon$. Answer: TRUE

*$\exists N\in\mathbb{Z^+},\forall\varepsilon>0,\ \ \text{whenever}\ n>N\Rightarrow|a_n-a|<\varepsilon$. Answer: FALSE 
I am confused that what is difference between 1 & 2 (or say why No.2 is wrong)? 
Thanks!

The first is the correct statement. It says: no matter how small $\epsilon$ is, you can always choose a large enough $N$ so that every $a_n$ is within $\epsilon$ of $a$ whenever $n>N$. The second statement says something quite different. It says that if I hand you some large integer $N$, then no matter how small $\epsilon$ is, $a_n$ is within $\epsilon$ of $a$ whenever $n>N$. This second statement is plainly false. For example, after I give you $N$, simply choose $\epsilon$ to be smaller than some difference $|a_n-a|$ for $n>N$.
