Thread: Help understanding proof

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

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.

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

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