Sequence

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

I was taught to denote the sequence [math]\{a_1,a_2,...\}[/math] by [math]\{a_n\}_{n=1}^\infty[/math] however I don't like this, as it looks like a set. I have seen the notation [math](a_n)_{n=1}^\infty[/math] 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, [math]f:\mathbb{N}\rightarrow S[/math] where [ilmath]S[/ilmath] is some set. For a finite sequence it is simply [math]f:\{1,...,n\}\rightarrow S[/math]. 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 a sequence:

• [ilmath]k:\mathbb{N}\rightarrow\mathbb{N} [/ilmath] which operates on an [ilmath]n\in\mathbb{N} [/ilmath] with [ilmath]n\mapsto k_n:=k(n)[/ilmath] where:
  • [ilmath]k_n[/ilmath] is increasing, that means [ilmath]k_n\le k_{n+1} [/ilmath]

We denote this:

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

See also

• Monotonic sequence
• Bolzano-Weierstrass theorem
• Cauchy sequence (Alternatively: Cauchy criterion for convergence)
• Convergence of a sequence (Or Limit (sequence) - the page Convergence of a sequence is being refactored into it)