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?

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:

$\left(\frac{1}{n}\right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

✅

The terms get within ±0.5 starting from

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

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.

$\left(\cos \left(\frac{n}{10}\right)^{10}\right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

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:

$\left(\frac{\cos (n)}{\sqrt n}\right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

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:

A sequence $(x_n)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$ converges to L if...

Empty space

Empty space

Empty space

Empty space

there is an *index*

where the sequence term at that index

beyond which all the sequence terms

is inside the tube

are inside the tube

for any tube centered at L, no matter how small

for a specific tube centered at L

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")

Often we make $\textcolor{#1d4ed8}{N}$ be the **very first** point where the terms enter the tube. But it doesn't have to be that way! $\textcolor{#1d4ed8}{N}$ can be **any** point beyond which all the terms are in the tube.

You're allowed to choose an extra big $\textcolor{#1d4ed8}{N}$ "just to be safe" if it makes you happy.

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

$\left(\frac{1}{n}\right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

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 anindexbeyond which all the sequence terms are inside the tube.

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

Tube terms | Math terms | Quantifiers |
---|---|---|

For any tube centered at L, no matter how small... | Empty space | Empty space |

there is an index... | Empty space | Empty space |

beyond which all the sequence terms... | Empty space | Empty space |

are inside the tube. | Empty space |

such that for all $\textcolor{#1d4ed8}{n \leq N}$

such that for all $\textcolor{#1d4ed8}{n \geq N}$

for all $\varepsilon < 0$

for all $\varepsilon > 0$

$|x_{\textcolor{#1d4ed8}{n}} - L| < \varepsilon$

$|x_{\textcolor{#1d4ed8}{n}} - L| > \varepsilon$

there exists some $\textcolor{#1d4ed8}{N \in \mathbb{N}}$

$\forall \varepsilon > 0$

$\forall \varepsilon < 0$

$\exists \varepsilon < 0$

$\exists \varepsilon > 0$

$\textcolor{#1d4ed8}{\forall n \leq N}$

$\textcolor{#1d4ed8}{\forall n \geq N}$

$\textcolor{#1d4ed8}{\exists n \leq N}$

$\textcolor{#1d4ed8}{\exists n \geq N}$

$\textcolor{#1d4ed8}{\forall N \in \mathbb{N}}$

$\textcolor{#1d4ed8}{\exists N \in \mathbb{N}}$

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.