Proof of the Fibonacci recurrence from the definition

Proof

Let $n$ be a natural number.
The node Fibonacci sequence defines $(F_m)_{m \ge 0}$ by $F_0 = 0$ , $F_1 = 1$ and, for every natural number $m$ , $F_{m+2} = F_{m+1} + F_m$ .
Taking $m = n$ in this defining rule gives $F_{n+2} = F_{n+1} + F_n$ , which is the claim.

Idea

The Fibonacci numbers are defined by their two starting values and this rule. So the recurrence holds by definition. Its use is that, with $F_0$ and $F_1$ , it fixes every later term.

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Fibonacci sequence example
Proof of the Fibonacci recurrence from the definition proof

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Tag	`B35K99ADCP`
URI	`m2://B35K99ADCP`
Kind	proof
Writing	W0 stub
MSC	11B39
Proves	Fibonacci recurrence
Uses	Fibonacci sequence
Changed	4 November 2026

Proof of the Fibonacci recurrence from the definition

Proof

Idea

Reading path

Where this leads

Further reading

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