The limit of sequence is the number L if for any segment with center L, all but a finite number of points are inside that segment.

In other words, for any ε > 0, if L is the limit of the sequence, all but a finite number of points are inside the segment (L-ε, L+ε).

## How to prove that L is limit of sequence?

What we need to prove to show that L is limit of sequence?

Let *ε*>0 be a real number, we need to show that all points but a finite number of points are inside the segment (L-ε, L+ε).

If all points but a finite number of points are inside the segment (L-ε, L+ε), there exist n_{0} so for each n>n_{0} , a_{n} is inside the segment (L-ε,L+ε). Can you explain why?

Suppose, there is no such n_{0}, for each n_{0} we choose there is at least one point outside the segment (L-ε,L+ε). In other words, there are infinite points outside the segment.

In other words, we need to show for each *ε*>0, there exist n_{0} for each n>n_{0} , a_{n} is inside the segment (L-ε,L+ε).

a_{n} inside the segment (L-ε,L+ε) if a_{n} > L-ε and a_{n} < L+ε. In other words, we can say that a_{n} inside the segment (L-ε,L+ε) if a_{n}-L>-ε and a_{n}-L<ε or if |a_{n}-L|<ε.

## Sample 1: The limit of sequence a_{n}=n/(n+1)

In this case, the limit of the sequence is 1. What does it mean?

For any segment with center 1, all but a finite number of points are inside that segment. Can you prove it?

## Sample 2: The limit of sequence a_{n}=(-1)^{n}/n

In this case, the limit of the sequence is 0. What does it mean?

For any segment with center 0, all but a finite number of points are inside that segment. Can you prove it?