## Number theory

Let $x_n$ be the n-th non-square positive integer. Thus $x_1=2, x_2=3, x_3=5, x_4=6,$ etc. For a positive real number x, denotes the integer closest to it by $\langle x\rangle$. If $x=m+0.5$, where m is an integer, then define $\langle x\rangle=m$. For example, $\langle 1.2\rangle =1, \langle 2.8 \rangle =3, \langle 3.5\rangle =3$. Show that $x_n=n+\langle \sqrt{n}\rangle$