Length of Fibonacci String is Fibonacci Number

Theorem
Let $S_n$ denote the $n$th Fibonacci string.

Let $\len \left({S_n}\right)$ denote the length of $S_n$.

Then:
 * $\len \left({S_n}\right) = F_n$

where $F_n$ denotes the $n$th Fibonacci number.

Proof
The proof proceeds by strong induction.

For all $n \in \Z_{>0}$, let $P \left({n}\right)$ be the proposition:
 * $\len \left({S_n}\right) = F_n$

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

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

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

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

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $P \left({j}\right)$ is true, for all $j$ such that $1 \le j \le k$, then it logically follows that $P \left({k + 1}\right)$ is true.

This is the induction hypothesis:
 * $\len \left({S_k}\right) = F_k$

and:
 * $\len \left({S_{k - 1} }\right) = F_{k - 1}$

from which it is to be shown that:
 * $\len \left({S_{k + 1} }\right) = F_{k + 1}$

Induction Step
This is the induction step:

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

Therefore:
 * $\forall n \in \Z_{>0}: \len \left({S_n}\right) = F_n$