22 Summary

This now concludes our section on real analysis. We examined some axioms and the topology of the real numbers. We saw how the Axiom of Completeness led to the incredible result that between any two real numbers, we can find both a rational and an irrational number. We also looked at open and closed sets and learned that every bounded, closed set is compact and vice versa. We then looked at sequences, limits, and continuity and saw how calculus and analysis work together within each of these areas. The ideas of analysis can definitely be confusing sometimes, but once they are understood, you can see an amazing world open up before your eyes.

We are now going to continue our study of sets by examining their underlying structures and patterns as we dive into the abstract world of modern algebra.