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The limit of a sequence as defined by my textbook:

Definition: Limit of a Sequence We say that $(a_n)_{n∈\mathbb{N}+}$ converges to limit L, and we write:

$\lim_{x\to 0}(a_n) = L$ or $(a_n)\to L$

If, for every $\epsilon>0$, there is a number $M$ such that $|(a_n) - L| < ε$ for all $n > M$

I feel like I have a decent understanding of the definition but the part about epsilon confuses me. What is epsilon?

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Treat epsilon as a range in which the values of your sequence lie when the sequence begins to converge. For example, take $\epsilon=0.01$. This means that the distance between all of the values in your sequence and the limit $l$ must be strictly less than $0.01$. As you take smaller values of $\epsilon$, your sequence begins to get closer and closer to the limit. i.e., the distance between the values of the sequence and the limit gets progressively smaller, so the sequence begins to converge.

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  • $\begingroup$ What is it trying to show in the last part of the definition by saying that there is a number M such that |An - L| < ε for all n > M? $\endgroup$
    – Idiot
    Nov 2 '16 at 19:26
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    $\begingroup$ @Andrew $|(a_n) - L|$ is the distance between $(a_n)$ and $L$. We want $\epsilon$ to be arbitrarily small. Therefore, $|(a_n) - L|<\epsilon$ means that we want to make the the distance between $(a_n)$ and the limit $L$ as small as possible. "For all $n>M$" means that the inequality holds for any arbitrarily large value of n. Remember, $M$ is dependent on $\epsilon$, so if we take $\epsilon$ to be smaller, we have to take $M$ to be bigger so that $|(a_n) - L|<\epsilon$ is still satisfied. $\endgroup$
    – aL_eX
    Nov 2 '16 at 21:42
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What trips many students up is the phrase "if, for any $\epsilon > 0$, ..." and that's normal. I'll explain it in pieces.

If, for every $\epsilon > 0$...

The $\epsilon$ here is given. It's whatever. We never know what it is. It's arbitrary. Really, it's a small number that defines a tolerance.

...there exists $M$ such that $|a_n - L| < \epsilon$ whenever $n > M$.

We must first determine what $M$ shall be. This is usually done by construction. The definition says whenever the index exceeds this number, all the terms will be within epsilon of the proposed limit $L$.

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