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Thread: delta-epsilon proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    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
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  2. #2
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    Quote Originally Posted by Danneedshelp View Post
    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.
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