Question 135
Given an integer $n \gt 1$, let the real number $x \gt 1$ satisfy
$$
x^{101} - n x^{100} + n x - 1 = 0.
$$
Prove that for any real numbers $0 \lt a \lt b \lt 1$, there exists a positive integer $m$ such that
$$
a \lt \{x^m\} \lt b.
$$
Here $\{t\} = t - \lfloor t \rfloor$ denotes the fractional part of the real number $t$.
$$
x^{101} - n x^{100} + n x - 1 = 0.
$$
Prove that for any real numbers $0 \lt a \lt b \lt 1$, there exists a positive integer $m$ such that
$$
a \lt \{x^m\} \lt b.
$$
Here $\{t\} = t - \lfloor t \rfloor$ denotes the fractional part of the real number $t$.