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Do you have an elementary proof for:

$$(\forall\epsilon>0)(\exists n,k\in\mathbb{N})(|\frac{n}{k}-\pi|<\frac{\epsilon}{k})$$

what is the largest $\delta>1$ such that:

$$(\forall\epsilon>0)(\exists n,k\in\mathbb{N})(|\frac{n}{k}-\pi|<\frac{\epsilon}{k^\delta})$$

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@Chris Eagle: real-analysis usually means measure theory. – user59671 Jan 30 '13 at 22:00
What? No it doesn't. – Chris Eagle Jan 30 '13 at 22:02
Do you mean the largest $\delta$ in your second question? – Rahul Jan 30 '13 at 22:24
The question seems to assume implicitly that there exists a largest $\delta$ such that the second statement is true. I feel that the possibility that the second statement is true for all $\delta$ is being excluded. – Amr Jan 30 '13 at 22:31
let's think supremum is the meaning. – user59671 Jan 30 '13 at 22:33
up vote 2 down vote accepted

I have a proof for the first one. Note that $\forall \epsilon>0\,\exists n\in\mathbb{N}\,\exists k\in \mathbb{Z}^+\,\left[\left|\frac{n}{k}-\pi\right|<\frac{\epsilon}{k}\right]$ is equivalent to: $$\forall \epsilon>0\,\exists n\in\mathbb{N}\,\exists k\in \mathbb{Z}^+\,[\left|n-k\pi\right|<\epsilon]$$

Let $\epsilon>0$. Let $10^{-r}<\epsilon $ for some $r\in\mathbb{N}$. Now consider the set of numbers $\{k\pi \mid k\in\mathbb{Z}\}$. By the pigeonhole principle, we know that there exist distinct $i,j\in\mathbb{N}$ $(i>j)$ such that $i\pi,j\pi$ have the same first $r$ digits that are on the right of the decimal point. Therefore, $|(i-j)\pi-\lfloor(i-j)\pi\rfloor|<10^{-r}<\epsilon$. Let $n=\lfloor(i-j)\pi)\rfloor$, $k=i-j$ and we get the desired result.

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by $\mathbb{N}$ I mean positive integers. – user59671 Jan 30 '13 at 22:38
I see. I use $\mathbb{N}$ for the set of natural numbers – Amr Jan 30 '13 at 22:40
great. accepted answer. what is $k>0$!? – user59671 Jan 30 '13 at 22:43
I removed $k>0$ – Amr Feb 5 '13 at 4:37

For the second question see mathworld

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