What is the technical definition of sequence convergence?

In the last lesson we said that a sequence converges if its terms get "infinitely close" to some limit value. But what exactly does that mean?

"As close as you want"

A sequence converges if it can get as close as you want to the limit value, just by scrolling along the sequence to a larger $\textcolor{#1d4ed8}{n}$ value.

For example, if you want $\left( \frac{1}{n} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$ to get within ±0.1 of its limit value, 0, you just have to scroll to $\textcolor{#1d4ed8}{n = 11}$. And if you want it to get within ±0.01 of its limit, you have to scroll to $\textcolor{#1d4ed8}{n = 101}$. No matter how close you want to get to the limit, there's a place where it happens.

We can visualize this by drawing a tube of closeness and waiting for the sequence terms to enter the tube. Try making this tube bigger or smaller and watch what happens:

No matter how small the tube gets, the terms of a converging sequence will eventually go inside.

You gotta stay in the tube

Here's an example of a sequence that—surprisingly—does NOT converge to 0 (it is diverging):

When you first look, it certainly appears to converge to 0! But in fact, as you scroll farther, you can see that it never stays in the tube. It always leaves eventually.

This is how we know that the sequence does NOT converge to 0.

A converging sequence, on the other hand, will become a permanent resident eventually. This one is undecided at first, but after a while it enters the tube permanently:

Turn it into a sentence

Alright... Time to describe, in a sentence, what it means to be convergent.

Below are some fridge magnets that you can use to build the definition. Drag them into place and then click the button to check your answer:

Make it mathy

Amazing! You've successfully built a definition of sequence convergence. But our current tube terminology—rad as it is—doesn't really lend itself to writing proofs.

Let's rewrite our definition to be more mathematical, starting by defining some variables:

• Call the tube radius $\varepsilon$ ("epsilon")
• Call the tube center point L. (This is the value we think the sequence is converging to.)
• Call the point where the terms enter the tube $\textcolor{#1d4ed8}{N}$ ("big n")

Here's the interactive tube graph with $\varepsilon$ and $\textcolor{#1d4ed8}{N}$ labeled:

Now that we have these two variables, we can write the definition of convergence more precisely. As a reminder, here's the definition in tube terms:

A sequence $(x_n)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$ converges to L if... for any tube centered at L, no matter how small, there is an index beyond which all the sequence terms are inside the tube.

Now, fill in the equivalent math terms using the fridge magnets below:

So our final definition is...

A sequence $(x_n)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$ converges to L if for all $\varepsilon > 0$, there exists some $\textcolor{#1d4ed8}{N \in \mathbb{N}}$ such that for all $\textcolor{#1d4ed8}{n \geq N}$, $|x_{\textcolor{#1d4ed8}{n}} - L| < \varepsilon$.

In the next lesson, we'll use this definition to prove that a specific sequence is convergent.