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Thread: Need help understanding "limit of a sequence" definition

  1. #1
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    Need help understanding "limit of a sequence" definition

    The picture of the definition is enclosed in the attachments:

    I don't understand what the whole greek letter epsilon and this introduced letter "M" mean.

    Please reply as if you're talking to a kindergartner because I really don't understand this complicated mathematical jargon.

    Thanks!
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  2. #2
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by RedSwiss View Post
    The picture of the definition is enclosed in the attachments:

    I don't understand what the whole greek letter epsilon and this introduced letter "M" mean.

    Please reply as if you're talking to a kindergartner because I really don't understand this complicated mathematical jargon.

    Thanks!
    that means that every number from sequence $\displaystyle a_n$ except of them finite many, are in $\displaystyle \varepsilon $ region of the number L. actually definition of the sequence...

    $\displaystyle \displaystyle (\forall \varepsilon >0) ( \exists n_0 \in \mathbb{N}) (\forall n \in \mathbb{N}) (n\ge n_0 \Rightarrow |x_n-a| < \varepsilon ) $

    this up there is same as if you write

    $\displaystyle \displaystyle \lim_{n\to \infty} x_n = a $

    if there is $\displaystyle a\in \mathbb{R}$, so that $\displaystyle \displaystyle \lim_{n\to \infty} x_n = a $ then we say that sequence $\displaystyle x_n$ converge.


    if you look at sequence $\displaystyle (-1) ^n$ you know that it's not converge (it diverges) but let's try to show that

    okay, let's assume that converges, and that there is for some $\displaystyle a\in \mathbb{R}$ we have that $\displaystyle \displaystyle \lim_{n\to \infty} x_n = a $. if we look at members of the sequence we see that all of them can be (1) or (-1), that means that both of those numbers have to be in any $\displaystyle \varepsilon $ region of the point a. but that is not possible if you chose that your $\displaystyle \varepsilon < \frac {1}{2}$ there is no way to both of those numbers to be in that interval $\displaystyle (a-\varepsilon , a+ \varepsilon )$ which length is less the one

    "we say that sequence $\displaystyle x_n$ converges to the point $\displaystyle a \in \mathbb{R}$ if in every $\displaystyle \varepsilon $ region of the point $\displaystyle a$ are almost all members of the sequence "
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