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 n\textcolor{#1d4ed8}{n} value.

Purple inflatable tube man

For example, if you want (1n)n∈N\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 n=11\textcolor{#1d4ed8}{n = 11}. And if you want it to get within ±0.01 of its limit, you have to scroll to n=101\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:

-1.5-1-0.500.511.5
n∈N\textcolor{blue}{n \in \mathbb{N}}
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(1n)n∈N\left(\frac{1}{n}\right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}
βœ…
The terms get within Β±0.5 starting from

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 thatβ€”surprisinglyβ€”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.

-0.200.20.40.60.811.2
n∈N\textcolor{blue}{n \in \mathbb{N}}
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(cos⁑(n10)10)n∈N\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:

-0.8-0.6-0.4-0.200.20.40.60.8
n∈N\textcolor{blue}{n \in \mathbb{N}}
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(cos⁑(n)n)n∈N\left(\frac{\cos (n)}{\sqrt n}\right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}

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:

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

Make it mathy

80s aesthetic text that says 'Totally Tubular Terminology'

Amazing! You've successfully built a definition of sequence convergence. But our current tube terminologyβ€”rad as it isβ€”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 Ξ΅\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 N\textcolor{#1d4ed8}{N} ("big n")
Side note

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

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

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

-1.5-1-0.500.511.5
N\textcolor{#1d4ed8}{N}
n∈N\textcolor{blue}{n \in \mathbb{N}}
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(1n)n∈N\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 (xn)n∈N(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:

Tube termsMath termsQuantifiers
For any tube centered at L, no matter how small...
there is an index...
beyond which all the sequence terms...
are inside the tube.
such that for all n≀N\textcolor{#1d4ed8}{n \leq N}
such that for all nβ‰₯N\textcolor{#1d4ed8}{n \geq N}
for all Ξ΅<0\varepsilon < 0
for all Ξ΅>0\varepsilon > 0
∣xnβˆ’L∣<Ξ΅|x_{\textcolor{#1d4ed8}{n}} - L| < \varepsilon
∣xnβˆ’L∣>Ξ΅|x_{\textcolor{#1d4ed8}{n}} - L| > \varepsilon
there exists some N∈N\textcolor{#1d4ed8}{N \in \mathbb{N}}
βˆ€Ξ΅>0\forall \varepsilon > 0
βˆ€Ξ΅<0\forall \varepsilon < 0
βˆƒΞ΅<0\exists \varepsilon < 0
βˆƒΞ΅>0\exists \varepsilon > 0
βˆ€n≀N\textcolor{#1d4ed8}{\forall n \leq N}
βˆ€nβ‰₯N\textcolor{#1d4ed8}{\forall n \geq N}
βˆƒn≀N\textcolor{#1d4ed8}{\exists n \leq N}
βˆƒnβ‰₯N\textcolor{#1d4ed8}{\exists n \geq N}
βˆ€N∈N\textcolor{#1d4ed8}{\forall N \in \mathbb{N}}
βˆƒN∈N\textcolor{#1d4ed8}{\exists N \in \mathbb{N}}

So our final definition is...

A sequence (xn)n∈N(x_n)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}} converges to L if for all Ξ΅>0\varepsilon > 0, there exists some N∈N\textcolor{#1d4ed8}{N \in \mathbb{N}} such that for all nβ‰₯N\textcolor{#1d4ed8}{n \geq N}, ∣xnβˆ’L∣<Ξ΅|x_{\textcolor{#1d4ed8}{n}} - L| < \varepsilon.

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