Number of Fibonacci Numbers between n and 2n

Theorem
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.

Then there exists either one or two Fibonacci numbers between $n$ and $2 n$ inclusive.

Proof
First existence is demonstrated.

Let $F_m \ge n$ such that $F_{m - 1} < n$.

This shows that the smallest Fibonacci number greater than $n$ is less than $2 n$.

Thus there exists at least one Fibonacci number between $n$ and $2 n$.

there exist $3$ Fibonacci numbers between $n$ and $2 n$.

Let $F_m \ge n$ be the smallest of those Fibonacci numbers.

Then:

But $F_{m + 2} < 2 n$ by hypothesis.

Hence by Proof by Contradiction there can be no more than $2$ Fibonacci numbers between $n$ and $2 n$.

Let $n = 2$.

Then between $2$ and $4$ there exist $F_3 = 2$ and $F_4 = 3$.

Let $n = 10$.

Then between $10$ and $20$ there exists $F_7 = 13$ and no other Fibonacci numbers.

Thus it has been demonstrated:


 * There always exists at least one Fibonacci number between $n$ and $2 n$


 * There never exist more than $2$ Fibonacci number between $n$ and $2 n$


 * There exist $n$ such that there exists exactly one Fibonacci number between $n$ and $2 n$


 * There exist $n$ such that there exist exactly $2$ Fibonacci numbers between $n$ and $2 n$.

The result is complete.