An in nite sequence of numbers is a function whose domain is the set of positive integers Z+ or natural numbers N. If a sequence has domain only the numbers, then the sequence is called a nite sequence.

While the idea of a sequence of numbers, a1, a2, a3, . . . is straightforward, it is useful to think of a sequence as a function. We have up until now dealt with functions whose domains are the real …

We have already seen that there is a sequence in a set S that converges to inf S and another that converges to sup S: In this section, we investigate other characteristics of sets and points that would …

real sequence is just a list of real numbers in order. If Υ is replaced with ≤, then we have a complex sequence. In this course, we will almost always deal with real sequences.

It is sometimes possible to assert that a sequence is convergent even if we can't nd it's limit directly. One way to do this it by using the least upper bound property of the real numbers.

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A Cauchy sequence of rational numbers converges to a unique real number, of course, and two sequences are equivalent if and only if they converge to the same real number.