Honsberger's Identity/Proof 1

Proof
From the initial definition of Fibonacci numbers, we have:
 * $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$

Proof by induction:

For all $n \in \Z_{>0}$, let $P \left({n}\right)$ be the proposition:
 * $\displaystyle \forall m \in \Z_{>0} : F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$

Basis for the Induction
$P \left({1}\right)$ is the case:

and so $P \left({1}\right)$ is seen to hold.

$P \left({2}\right)$ is the case:

and so $P \left({2}\right)$ is seen to hold.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ and $P \left({k-1}\right)$ are true, where $k > 1$, then it logically follows that $P \left({k + 1}\right)$ is true.

So this is our induction hypothesis:
 * $\displaystyle F_{m + k} = F_{m - 1} F_k + F_m F_{k + 1}$

and:
 * $\displaystyle F_{m + k - 1} = F_{m - 1} F_{k - 1} + F_m F_k$

from which it is to be shown:
 * $\displaystyle F_{m + k + 1} = F_{m - 1} F_{k + 1} + F_m F_{k + 2}$

Induction Step
This is our induction step:

So $P \left({k}\right) \land P \left({k - 1}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall m, n \in \Z_{>0} : F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$