Question 5
For any positive integer $n$, let $f(n)$ denote the $n$th positive nonsquare integer, i.e., $f(1)=2$, $f(2)=3$, $f(3)=5$, $f(4)=6$, etc. Prove that
$$f(n)=n+\{\sqrt n\},$$
where $\{x\}$ denotes the integer closest to $x$. (For example, $\{\sqrt1\}=1$, $\{\sqrt2\}=1$, $\{\sqrt3\}=2$, $\{\sqrt4\}=2$.)
$$f(n)=n+\{\sqrt n\},$$
where $\{x\}$ denotes the integer closest to $x$. (For example, $\{\sqrt1\}=1$, $\{\sqrt2\}=1$, $\{\sqrt3\}=2$, $\{\sqrt4\}=2$.)