N(E) means a natural number dependent on E.

Consequently, trying to understand it practically is simple.

is a sequence given to you.

What this definition essentially claims is:

If given any positive real number(generally we denote it by E) to you, then can you find a tail of the sequence for which all the numbers of the tail are less than that E.

Say I give you , my job is to find a tail for your sequence. I say "hey look, I notice that . Oh and that is the 10000000th term in your sequence. And from there every term is less than this number. In other words, the tail of your sequence starting from the 10000000th term satisfies the definition."

So one such .

Two things noteworthy here:

1) 10000000 is not at all special.

2) And I have proved it for one E, how do we prove this for every E.

For 2, here is the answer.

Notice this line in the proof:

Why do you think you can do that? If you can tell me this I will continue with my explanation, if you need it.E>0 given

take N(E) s.t. N(E) > 1/E