Calculate the $\sup$

Question

Let us $A=\{\frac{1}{|\frac{\pi}{2}+2n\pi|}, n\in\mathbb Z \}$, Calculate $\sup A$.

I am confuse, because $n\in\mathbb Z$ then for me i need to find the infimum of $|\frac{\pi}{2}+2n\pi|$ that it goes to $\infty$ implies that $\sup A= 0$ right?

I thought well? or i am wrong?

Thank you

Answer 1

Notice that $$ \frac{1}{\left| \frac{\pi}{2} + 2\pi n \right|} = \frac{2}{\left| \pi + 4\pi n \right|} = \frac{2}{\pi} \cdot \frac{1}{\left|4n+1\right|} $$ So if you can find the supremum of $\left\{\left|\frac{1}{4n+1}\right|\right\}$, you are finished.

Answer 2

The infimum is $\frac{\pi}{2}$ because $\lvert \frac{\pi}{2}+2n\pi\rvert \geq\frac{\pi}{2} $ and it is achieved for $n=0$, so $\sup A=\frac{2}{\pi}$ and $\inf A=0$