Well, this is it. After handing in the final draft of my research project’s report, I’ve officially finished my second year of university. At McGill, science students spend their first year fulfilling freshman requirements: the standard set of chemistry, physics, and calculus. Your first year (or U0 as we call it here) is where you try out several subjects and see which strikes your fancy. Second year is when you start to specialize. It’s in second year that math students are treated to their first real proof-based courses: some mix of analysis, algebra, advanced calculus and probability theory. The content of these courses isn’t in itself particularly sophisticated; rather, the function of 200-level classes is introduce the student to fundamental concepts of mathematical rigor and the structure of arguments while also laying the foundations of the subjects for later study.
There were still some pretty nifty proofs that turned up this year though, so I thought I’d share a couple of them here.
The proof that the rationals have the same cardinality as the natural numbers
The dual of the dual of a vector space is isomorphic to that vector space
The weak law of large numbers and the central limit theorem
There exists no translational-invariant measure that can be defined on all subsets of \(\mathbb{R}\).
The Lebesque criterion for Riemann integrability
For the moment the curious reader can google these, but I’m planning on adding a blog post for each of these proofs later. Till then, happy summer!