ABSTRACT
Sequences are the nuts and bolts of real analysis. All the basic ideas are formulated in terms of sequences. A subsequence of a given sequence is a "junior" sequence that lives inside the parent sequence. Many of the most important ideas in analysis, including closure and compactness, are formulated in terms of subsequences. A subsequence of a given sequence is also a subset of that sequence. But it is much more than that, because it maintains the same order of terms. Any convergent sequence is Cauchy. And, in the real number system, any Cauchy sequence is convergent. The answer is easy. Convergent sequences are what we are really interested in. But the concept of Cauchy sequence helps us to identify them. The Bolzano-Weierstrass theorem is a generalization of our result from the last section about increasing sequences which are bounded above. Convergent sequences are useful objects, but the unfortunate truth is that most sequences do not converge.
