For $a,binmathbb{R}$, there is an integer within $|{a} - {b}|$ from $|a-b|.$

Mathematics Asked by MathGeek on December 26, 2020

Let’s take two real numbers $a,b$. The distance between $a$ and $b$ is $|a-b|$. Let ${}$ denote fractional part. Then for any $a$ and $b$, there is an integer close to $|a-b|$ which is at most $|{a} – {b}|$ away from $|a-b|.$

We can see this intuitively but I just want to know what proof technique can be used and how to prove the above statement?

One Answer

We may assume that $a leq b$, so that $| a - b | = b-a$. Consider $n = (b - {b}) - (a - {a})$. Then $n$ is an integer and we have $$|n - |a - b|| = |n - b + a| = |{a} - {b}|,$$ as desired.

Correct answer by Dániel G. on December 26, 2020

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