An Introductory Proof

This document shows how to write theorems and proofs, The specific example is a theorem about rational numbers:

Theorem 1. If \(q\) is a rational number and \(n\) is an integer, then \(q+n\) is rational.

Proof. We assume that \(q\) is a rational number and \(n\) is an integer, and will show that \(q+n\) must be rational. Since \(q\) is rational, we can write \(q = \frac {a}{b}\) where \(a\) and \(b\) are integers and \(b \ne 0\). We also observe that \(n = \frac {nb}{b}\). Therefore

\begin{equation*} q + n = \frac {a + bn}{b} \end{equation*}

Since integers are closed under addition and multiplication, \(a + bn\) is an integer. Furthermore, \(b\) is a non-zero integer, and so \(\frac {a+bn}{b}\) is a rational number. We have now shown that if \(q\) is a rational number and \(n\) is an integer, that \(q+n\) is rational. □