Lemma 2.2.2. For any natural number \(n\), \(n+0=n\).
\(Proof~by~Tao\). We use induction. The base case \(0+0=0\) follows since we know that \(0+m=m\) for every natural number \(m\), and \(0\) is a natural number. Now suppose inductively that \(n+0=n\). We wish to show that \((n\pp)+0=n\pp\). But by definition of addition, \((n\pp)+0\) is equal to \((n+0)\pp\), which is equal to \(n\pp\) since \(n+0=n\). This closes the induction. \(\square\)