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

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?