## Cantor set applet

I made this Cantor set applet for my Real Analysis class. It is nothing fancy, but it saves me from drawing it on the board.

## Applet to illustrate the epsilon-delta definition of limit

Here’s a GeoGebra applet that I made for my Real Analysis class. It can be used to explore the definition of limit: Definition. The limit of as approaches is , or equivalently if for any there exists such that whenever , it follows that .

## An applet for teaching the limit of a sequence

I’m currently teaching real analysis. Right now we’re discussing limits of sequences. The definition is: The limit of a sequence is (or converges to ) if, given any , there exists a natural number such that for all . I used GeoGebra to create the following applet, which illustrates the definition of a limit. (Clicking…

## Cardinality of infinite sets, part 1: four nonstandard proofs of countability

The study of cardinalities of infinite sets is one of the most intriguing areas of mathematics that an undergraduate mathematics major will encounter. It never fails to bring crooked smiles of joy, disbelief, confusion and wonder to their faces. The results are beautiful, deep, and unexpected. Recall that two sets have the same cardinality if…

## Google Books replaces the index

Indexes can be great tools for finding specific information in books. However, as we all know, they are often maddeningly incomplete. I was constantly frustrated when I was in graduate school studying for my analysis prelim exam. Royden’s Real Analysis (our text) had a terrible index. I ended up hand writing dozens of entries into…

## Kuratowski’s closure-complement theorem (solution)

Stop!  This post contains spoilers.  This page has the solution to the problem posed in yesterday’s post. We challenged you to find a set from which we can make as many new sets as possible using only the closure and complement operations. In 1922 Kuratowski proved the following theorem. Theorem. At most 14 sets can…

## Kuratowski’s closure-complement theorem

One of my favorite theorems in elementary topology is Kuratowski’s closure-complement theorem. First some notation.  For any set let denote the complement of and  denote the closure of .  (Recall that and  is the union of and all the limit points of ). Here’s the problem.  Find a set so that we can construct as many…