Help understanding proof

**zebra2147** · November 9th 2010, 05:03 AM

I came across a proof in my notes that I will be more then happy to try on my own but I don't really know what for sure they are looking for. If anyone could just point me in the right direction I would appreciate it.

Let $a$ be some irrational number. Given any $M\in \mathbb{N}$ , there is a $\gamma >0$ , such that for all $p\in \mathbb{Z}$ and $q\in \mathbb{N}, q\leq M\Rightarrow |a-p/q|\geq \gamma$ . [Hint: There are only finitely many $p$ such that $a-\gamma \leq p/q\leq a+\gamma$ .]

Like I said, I just don't know where to get started. All we have talked about up to this point in the chapter was the basics continuity, composition rule, and removable discontinuities. Any help would be appreciated.

**Opalg** · November 9th 2010, 10:58 AM

Originally Posted by zebra2147

I came across a proof in my notes that I will be more then happy to try on my own but I don't really know what for sure they are looking for. If anyone could just point me in the right direction I would appreciate it.

Let $a$ be some irrational number. Given any $M\in \mathbb{N}$ , there is a $\gamma >0$ , such that for all $p\in \mathbb{Z}$ and $q\in \mathbb{N}, q\leq M\Rightarrow |a-p/q|\geq \gamma$ . [Hint: There are only finitely many $p$ such that $a-\gamma \leq p/q\leq a+\gamma$ .]

Like I said, I just don't know where to get started. All we have talked about up to this point in the chapter was the basics continuity, composition rule, and removable discontinuities. Any help would be appreciated.

There are only finitely many values of $q$ from 1 to $M$ (obviously). For each of these, look at the multiple of 1/ $q$ that is closest to $a$ . Let $d_q$ be the distance of this point from $a$ . Then choose $\gamma$ so that $0<\gamma<\min\{d_q:1\leqslant q\leqslant M\}$ , and explain why that $\gamma$ has the required property.

Math Help - Help understanding proof

LinkBack

Thread Tools

Display

Help understanding proof

Similar Math Help Forum Discussions

Understanding Topology Proof

Not Understanding Proof for Derivative.

Need help understanding proof of theorem

Help Understanding a Proof

Understanding logic of proof

Search Tags