On Getting Comfortable With Infinity
I'm over eight weeks into my Analysis class here at school. While I certainly like the applied side of mathematics more than the pure side, I've found my Analysis class to be exceptionally interesting and thought provoking. I enjoy the act of thinking very deeply about mathematical concepts that we normally take for granted such as the existence of the real numbers and whether a set is open or closed. Another idea that has struck me is how infinity seems to lie somewhere in every definition, from the real numbers to compact sets. Like any Analysis class we spend a lot of time proving theorems and properties surrounding these definitions, which means working a lot with infinity. It seems as if I'm always on one of two sides; using infinity to prove something, or wrestling against infinity to prove something. To prove a sequence is Cauchy I normally take advantage of some characteristic the sequence exhibits from term to term for infinitely many terms. On the other hand to prove a set is compact I'm wrestling with an arbitrary open cover of infinite sets and laboring to reduce it to some finite subset that still covers our set. Sometimes, as with countable and uncountable sets, I'm just trying to define the magnitude of infinity I'm dealing with.
Not only is Analysis teaching me the fundamentals of Calculus, but it is pushing me to become comfortable working with infinity. The latter is arguably just as important as the former.
Infinity is an immensely powerful concept. Though we can't really grasp the true notion of it, we can still somehow apply it to obtain the basis of so much of the mathematics we know and rely on today.