Question 17
Ambra constructs a sequence starting from a positive real number $x_0$ as follows. Given the term $x_n$, she sets
$$
x_{n+1}=\left\{1-\frac1{x_n}\right\},
$$
where $\{\alpha\}$ is the fractional part of $\alpha$, the difference between $\alpha$ and the greatest integer less than or equal to it. As soon as she obtains a term equal to zero, she stops the sequence.
Part a: Prove that if $x_0=p/q$ for positive integers $p,q$, then the sequence eventually stops.
Part b: Prove that if the sequence stops, then its starting value $x_0$ was of the form $p/q$ for positive integers $p,q$.
$$
x_{n+1}=\left\{1-\frac1{x_n}\right\},
$$
where $\{\alpha\}$ is the fractional part of $\alpha$, the difference between $\alpha$ and the greatest integer less than or equal to it. As soon as she obtains a term equal to zero, she stops the sequence.
Part a: Prove that if $x_0=p/q$ for positive integers $p,q$, then the sequence eventually stops.
Part b: Prove that if the sequence stops, then its starting value $x_0$ was of the form $p/q$ for positive integers $p,q$.