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Thread: Definition of a limit of a sequence

  1. #1
    Feb 2013

    Definition of a limit of a sequence

    Hi all,

    Can anyone help me to understand the definition of a limit of a sequence. In notes and when reading about it on the internet, I always find for every ε > 0 ∃ n such that for every n ≥ N, mod(An - L) < ε.

    My biggest problem is picturing this. I understand that epsilon is a value very close to the limit of the sequence but I don't understand how this all relates together to allow us to show that a sequence converges to its limit e.g. 1/n^1/2 -> 0.

    Anybody able to explain this?

    Thanks guys
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  2. #2
    Senior Member
    Jan 2009

    Re: Definition of a limit of a sequence

    Let's take your example of the sequence $\displaystyle a_n = \frac{1}{\sqrt{n}}$

    The goal is to find an equation for N dependent on epsilon such that for $\displaystyle n \geq N$ $\displaystyle |a_n-L| < \epsilon$ If $\displaystyle L=0$, then

    $\displaystyle |a_n-L| = |\frac{1}{\sqrt{n}}| = \frac{1}{\sqrt{n}} < \epsilon \iff n > \frac{1}{{\epsilon}^2} = N}$

    so no matter how small epsilon (the interval) gets, we can always find a natural number $\displaystyle n \geq N$ such that a point $\displaystyle a_n$ is in that interval. As epsilon goes to 0, the difference between the points $\displaystyle a_n$ and $\displaystyle L$ becomes sufficiently small
    Last edited by MacstersUndead; Mar 10th 2013 at 05:59 PM.
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