Finding $\sup(a_n)$ and $\inf(a_n)$

Question

I was trying to figure out $\inf(a_n)$ and $\sup(a_n)$ of the sequence defined by $a_n=\arctan(n^2-7n+13)$. Since $a_n$ is increasing for $n \geq 4$, and $\sup(a_n)$ cannot be $a_1, a_2, a_3$, I thought it was possible to apply the monotone theorem which states $\lim\limits_{n \rightarrow +\infty}a_n=\sup(a_n)$, so $\sup(a_n)$ should be $\pi/2$. Is it correct? What should I do to find $\inf(a_n)$?

Answer 1

Yes, that is correct: $\sup\{a_n\mid n\in\Bbb N\}=\frac\pi2$ for that reason.

And $\inf\{a_n\mid n\in\Bbb N\}=\arctan(1)=\frac\pi4$, since $1$ is the smallest number of the form $n^2-7n+13$ ($n\in\Bbb N$). You can deduce this from the fact that$$n^2-7n+13=\left(n-\frac72\right)^2+\frac34$$and that $1=\left(\pm\frac12\right)^2+\frac34$.