Diagonalization
Definition. Diagonalization defeats any proposed infinite list by building an object that differs from the -th entry at the -th position, and therefore equals no entry of the list. Because the list was arbitrary, no list can work.
It proves the infinite binary sequences Uncountable (Lecture 2, Example 4), makes the halting problem undecidable (Lecture 7), and, beyond this course, separates complexity classes outright through the hierarchy theorems.