Question 95
A rational number $x$ is given. Prove that there exists a sequence $x_0, x_1, x_2, \dots$ of rational numbers with the following properties:
a. $x_0 = x$;
b. for every $n \ge 1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \frac{1}{n}$;
c. $x_n$ is an integer for some $n$.
(This problem was suggested by Gabriel Carroll.)
a. $x_0 = x$;
b. for every $n \ge 1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \frac{1}{n}$;
c. $x_n$ is an integer for some $n$.
(This problem was suggested by Gabriel Carroll.)