Questions about the definition of convergence

Question

I am having difficulty understanding the definition of convergence. I've been rereading and looking at examples during this past week and I haven't made any progress.

Definition: We say that {${a_{n}}$} converges to a point $a \in \mathbb{R}$ if for any $\epsilon$, there exists a positive integer $N$ such that for any $n \in \mathbb{N}$ with $n\geq N$, one has $|a_{n}-a|< \epsilon$.

My questions:

(1) What role does $\epsilon$ play? Is that the actual limit? I thought that $a$ is what we are "assuming" is the limit?

(2) Why does $n \geq N$? I'm asking this because we were just given the definition of what it means to be convergent (in Real Analysis, not the Calculus sequence), no formal proof.

(3) In various examples they are trying to set $N$ to be less than or equal to $\epsilon$. Why?

My main issue is that I don't understand how the components of this definition work. I can follow the examples, but I'd rather understand why it works then just take it on blind faith.

Thank you for any input/suggestions.

Answer 1

The limit is $a$, not $\varepsilon$.

The point is that one can make $a_n$ as close to $a$ as desired by making $n$ big enough. The absolute value $|a_n-a|$ is the distance between $a_n$ and $a$.

How big is big enough depends on how close you want to make $a_n$ to $a$.

So $\varepsilon$ is how close you want to make $a_n$ to $a$, and $N$ is how big you need to make $n$, i.e. as long as $n$ is $N$ or bigger, then $a_n$ is close enough to $a$.

The definition says that no matter how small $\varepsilon$ gets (as long as it's positive), $N$ can still be made big enough.

The suggestion that $N$ would be made equal to $\varepsilon$ is silly and makes me wonder if you were reading something about the limit of a function of a real variable rather than about the limit of a sequence. Generally $N$ will be something that depends on $\varepsilon$ and will get bigger as $\varepsilon$ gets smaller.

Answer 2

The idea of convergence for a sequence $\{a_n\}$ is more or less the following:

we say that $a_n$ converges to $a$ if for $n$ very very large, $a_n$ is very very close to $a$.

How do we translate this in math?

We use $\epsilon$ to say "how close" to $a$ we want to be, and $N$ to say "how far" in the sequence we need to go to stay that close.

So $\epsilon$ is to be thought to be a very small number, and $N$ a very large number. We try to read the definition again:

For every $\epsilon$ (in particular for every $\epsilon$ very small), there exists an $N$ (probably very large) such that, if the index $n$ of the sequence is greater than $N$, then the distance between the $n$-th term $a_n$ and $a$ is less than $\epsilon$.

To answer your questions:

(1) The limit is $a$.

(2) We say $n > N$ because the convergence only cares of what happens for large indices, where large determined by this parameter $N$, dependent on $\epsilon$.

(3) Usually in examples, given a generic $\epsilon$, we try to determine $N$ (as a function of $\epsilon$) such that the inequality $\vert a_n - a\vert < \epsilon$ holds for every $n > N$. But $\epsilon$ and $N$ live in two different worlds: $\epsilon$ lives in the world of the values of the sequence, $N$ lives in the world of the indices of the sequence.