Simple Proof Methods

A corollary is a result that follows directly from a theorem with little or no additional proof.

Example:

Corollary. [Square of an odd integer] If $n$ is an odd integer, then $n^2$ is also odd.

Proof. Let $n$ be an odd integer. According to the definition of odd integers, there exists $k \in \mathbb{Z}$ such that $n = 2k + 1$.
Notice that $$n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1,$$which is, by the definition of odd integers, odd because $(2k^2 + 2k)$ is an integer.◼

In $\LaTeX$, corollaries are written inside the

corollary

environment; see slide 36 for the full $\LaTeX$ code.