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

The fact that it goes to (positive) infinity doesn't really tell you much about its infimum, does it? — JonathanZ, Nov 01 '22 at 17:35
To find sup A you decided to look at the inf of the denominator, right? Noting that it goes to infinity, which you did correctly, tells you about its sup, not its inf. You've gotten your infs and sups mixed up. — JonathanZ, Nov 01 '22 at 17:41

score 1 · Accepted Answer · answered Nov 01 '22 at 17:40

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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.

answered Nov 01 '22 at 17:40

Matthew Leingang

26,112

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Thank you @Matthew Leingang, for me $\frac{1}{4n+1}$ is decreasing so the bigger value is fill in $n=0$ so $\sup \frac{1}{4n+1}=1$, – weymar andres Nov 01 '22 at 17:50

score 1 · Answer 2 · answered Nov 01 '22 at 17:48

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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 $\inf A=0$

answered Nov 01 '22 at 17:48

belkacem abderrahmane

2,801
5
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thank you so much, you confirm the above statement. – weymar andres Nov 01 '22 at 17:56

Calculate the $\sup$

2 Answers2