I am self-studying Terence Tao. I am stuck on the following question.
Exercise 4.4.1. Let $x$ be a rational number. Then there exists an integer n such that $n \leq x x < n + 1$. In fact, this integer is unique. I solved the uniqueness part.
For existence I have the following starting point. I am stuck though. let $x = \frac{p}{q}$.
By Euclidean algorithm there exists c and s such that $x = ps + c$ where $0 \leq c < s$.
I am not sure what to do after that.
Best Answer
Write $xx = \frac{p}{q}$ where $p \in \mathbb{Z}$, $q \in \mathbb{Z} > 0$. By the Euclidean algorithm, we can write $p = nq + s$ where $0 \leq s < q$. Hence $$nq \leq p < (n + 1)q.$$ This means $n \leq \frac{p}{q} < n + 1$.