# Sequence

A sequence is one of the earliest and easiest definitions encountered, but I will restate it.

I was taught to denote the sequence $\{a_1,a_2,...\}$ by $\{a_n\}_{n=1}^\infty$ however I don't like this, as it looks like a set. I have seen the notation $(a_n)_{n=1}^\infty$ and I must say I prefer it. This notation is inline with that of a tuple which is a generalisation of an ordered pair.

## Definition

Formally a sequence [ilmath](A_i)_{i=1}^\infty[/ilmath] is a function, $f:\mathbb{N}\rightarrow S$ where [ilmath]S[/ilmath] is some set. For a finite sequence it is simply $f:\{1,...,n\}\rightarrow S$. Now we can write:

• [ilmath]f(i):=A_i[/ilmath]

This naturally then generalises to indexing sets

## Notation

To specify that the points of a sequence, the [ilmath]x_i[/ilmath] are from a space, [ilmath]X[/ilmath] we may write:

• [ilmath](x_n)^\infty_{n=1}\subseteq X[/ilmath]

This is an abuse of notation, as [ilmath](x_n)^\infty_{n=1}[/ilmath] is not a subset of [ilmath]X[/ilmath]. It plays on:

• [ilmath][(x_n)^\infty_{n=1}\subseteq X]\iff[x\in(x_n)_{n=1}^\infty\implies x\in X][/ilmath]

Note that the elements of [ilmath](x_n)_{n=1}^\infty[/ilmath] are ether:

• Elements of a relation (if we consider the sequence as a mapping) or
• So using this, [ilmath]x\in(x_n)_{n=1}^\infty[/ilmath] may look like [ilmath]x=(a,b)[/ilmath] (indicating [ilmath]f(a)=b[/ilmath]) which is an Ordered pair, not in [ilmath]X[/ilmath]
• Elements of a tuple (which is a generalisation of ordered pairs where (usually) [ilmath](a,b)=\{\{a\},\{a,b\}\}[/ilmath]
• So using this, [ilmath]x\in(x_n)_{n=1}^\infty[/ilmath] may indeed look like [ilmath]x=\{\{a\},\{a,b\}\}\notin X[/ilmath]

As such the notation [ilmath](x_n)^\infty_{n=1}\subseteq X[/ilmath] having no other sensible meaning is a notation to say that [ilmath]\forall i[x_i\in X][/ilmath]

## Subsequence

Given a sequence [ilmath](x_n)_{n=1}^\infty[/ilmath] we define a subsequence of [ilmath](x_n)^\infty_{n=1}[/ilmath] as follows:

• Given any strictly increasing monotonic sequence[Note 1], [ilmath](k_n)_{n=1}^\infty\subseteq\mathbb{N}[/ilmath]
• That means that [ilmath]\forall n\in\mathbb{N}[k_n<k_{n+1}][/ilmath][Note 2]

Then the subsequence of [ilmath](x_n)[/ilmath] given by [ilmath](k_n)[/ilmath] is:

• [ilmath](x_{k_n})_{n=1}^\infty[/ilmath], the sequence whose terms are: [ilmath]x_{k_1},x_{k_2},\ldots,x_{k_n},\ldots[/ilmath]
• That is to say the [ilmath]i[/ilmath]th element of [ilmath](x_{k_n})[/ilmath] is the [ilmath]k_i[/ilmath]th element of [ilmath](x_n)[/ilmath]

### As a mapping

Consider an (injective) mapping: [ilmath]k:\mathbb{N}\rightarrow\mathbb{N} [/ilmath] with the property that:

• [ilmath]\forall a,b\in\mathbb{N}[a<b\implies k(a)<k(b)][/ilmath]

This defines a sequence, [ilmath](k_n)_{n=1}^\infty[/ilmath] given by [ilmath]k_n:= k(n)[/ilmath]

• Now [ilmath](x_{k_n})_{n=1}^\infty[/ilmath] is a subsequence