We had no homework assignments due on Thursday, so I dove right into lecture. We proved a few more results following from our ordering of the reals. Upon completing these theorems, we were then able to do the basic sorts of solving of inequalities we teach in our lowest level of algebra. It is interesting that we get to one of our final senior undergraduate courses and start doing things we cover in the earliest mathematics courses, but this time, we’ve built our understanding of these rules from the ground up. Have no fear, analysis students, we will go much, much beyond those mathematical tools we used intermediate algebra.
Next, we defined the concept of absolute value and some simple results of that definition. The next big thing after that was to use absolute values to create 
-neighborhoods which will provide us the necessary structures to define open and closed sets, cluster points, limits, etc.
Before we get to those concepts, however, we will divert to another low-level algebra topic of solving absolute value equations and inequalities. That will happen next time at the very beginning of class.
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