Yesterday at 6:30pm, I turned in my real analysis final. I am now (in theory) mathematically competent enough to do econ. Whether that is actually the case remains an open question…
Over the course of the semester, we learned dozens of definitions and theorems. However, one theorem is a workhorse that got me through the course: The Stack Exchange Approximation Theorem.
We start with two definitions.
Definition 1 (Error Function): Let S be a set of answers. Define the error function, g, by g: S -> N
The error function is a family of functions that counts the number of places a solution goes wrong.
Definition 2: A sequence of solutions converges to the correct solution if, for all e> 0, for some n*, n> n* => |g(s_n) – 100| <e.
The intuition behind definition 2 is that as solutions get closer to getting full marks, the more similar the answers are.
It is tempting to extrapolate this into a general convergence. that is
a solution converges if, for all e>0, for some n*, n > n* => |g(s_n) – g(s)| <e. However, the “Anna Karenina principle” applies. Every correct solutions are the same. Every incorrect solution is incorrect in its own way.
The Stack Exchange approximation theorem: Let f be a problem at the undergraduate level in Maths . Then there exists a series of Stack exchange questions that converge to the true solution to the problem.
Corollary (Generalisation): Let y be a problem at the undergraduate level in a STEM field . Then there exists a series of Stack exchange questions that converge to the true solution to the problem.
Proof of the theorem and corollary is left as an exercise to the reader.