Definition

An infinite sequence of numbers converges or  approaches alimit z, if eventually all its terms are near z. How near? As near as you want. 

The technical definition of convergence is: A sequence of numbers converges to a limit z if for any positive criterion q there is an n(q) so that every term in the  sequence after the n(q) th is within distance q of z .

Thus, for example, 1, 1/2, 1/3, ... 1/n, converges to 0; no matter how small q is, if  q > 0 then all terms after the [1 / q] thare less than q, and so within distance q of 0.   

A sequence that converges to z is sometimes said to "approach z" or "tend to z" as well.

We consider the sequence 1, 1/2, 1/3...

Suppose we choose q = 0.1

All the terms after the 10th are less than q = 0.1, so within distance q of z = 0.

A sequence s is said to converge if for any positive criterion q,all the terms in s beyond the n(q) th are within q of one another.

Definition

An interval of the real line is a set consisting of all real numbers between two endpoints, say, a and b . The interval is said to be closed if it includes the endpoints; 

open if it excludes them. We denote such intervals by [a,b] and ]a,b[ respectively.

 Definition

A set S is said to be complete if every sequence of members of S that converge, converges to some member of S . Closed intervals of the real line are complete; are open intervals?