delta-epsilon proof

• Oct 29th 2009, 10:00 PM
Danneedshelp
delta-epsilon proof
Q: Prove $\displaystyle lim_{x\rightarrow\pi}[[x]]=3$, where $\displaystyle [[x]]$ denotes the greatest integer less that or equal to x.

A: Here is some of my scratch work:

Let $\displaystyle \epsilon>0$. Suppse,

$\displaystyle 0<|x-\pi|<\delta=\frac{1}{10}$

$\displaystyle |x-\pi|<\delta=\frac{1}{10} \Leftrightarrow\frac{-1}{10}<x-\pi<\frac{1}{10} \Leftrightarrow\\3<x<4$

$\displaystyle \Rightarrow[[x]]=3$

$\displaystyle \Rightarrow|[[x]]-3|=|3-3|=0<\epsilon$

That is my attempt. Am I on the right path or is there more to show?

Thanks
• Oct 30th 2009, 03:28 AM
Plato
Quote:

Originally Posted by Danneedshelp
Q: Prove $\displaystyle lim_{x\rightarrow\pi}[[x]]=3$, where $\displaystyle [[x]]$ denotes the greatest integer less that or equal to x.

A: Here is some of my scratch work:

Let $\displaystyle \epsilon>0$. Suppse,

$\displaystyle 0<|x-\pi|<\delta=\frac{1}{10}$

$\displaystyle |x-\pi|<\delta=\frac{1}{10} \Leftrightarrow\frac{-1}{10}<x-\pi<\frac{1}{10} \Leftrightarrow\\3<x<4$

$\displaystyle \Rightarrow[[x]]=3$

$\displaystyle \Rightarrow|[[x]]-3|=|3-3|=0<\epsilon$

That is my attempt. Am I on the right path or is there more to show?

No, it is correct.