Editing Mathematical induction (section)

==== Example: Fibonacci numbers ====
Complete induction is most useful when several instances of the inductive hypothesis are required for each induction step. For example, complete induction can be used to show that
<math display="block"> F_n = \frac{\varphi^n - \psi^n}{\varphi - \psi}</math>
where <math>F_n</math> is the {{mvar|n}}-th [[Fibonacci number]], and <math display="inline">\varphi = \frac{1}{2}(1 + \sqrt 5)</math> (the [[golden ratio]]) and <math display="inline">\psi = \frac{1}{2} (1 - \sqrt 5)</math> are the [[root of a polynomial|roots]] of the [[polynomial]] <math>x^2-x-1</math>. By using the fact that <math>F_{n+2} = F_{n+1} + F_{n}</math> for each <math>n \in \mathbb{N}</math>, the identity above can be verified by direct calculation for <math display="inline">F_{n+2}</math> if one assumes that it already holds for both <math display="inline">F_{n+1}</math> and <math display="inline">F_n</math>. To complete the proof, the identity must be verified in the two base cases: <math>n = 0</math> and <math display="inline">n = 1</math>.