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 n₀ so for each n>n₀ , a_n is inside the segment (L-ε,L+ε). Can you explain why?

Suppose, there is no such n₀, for each n₀ 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 n₀ for each n>n₀ , 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?

The Samples