# delta-epsilon proof

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

A: Here is some of my scratch work:

Let $\epsilon>0$. Suppse,

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

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

$\Rightarrow[[x]]=3$

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

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

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

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

A: Here is some of my scratch work:

Let $\epsilon>0$. Suppse,

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

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

$\Rightarrow[[x]]=3$

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

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

No, it is correct.