Subsequence/Definition

Definition

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

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

The sequence:

[ilmath](x_{k_n})_{n=1}^\infty[/ilmath] (which is [ilmath]x_{k_1},x_{k_2},\ldots x_{k_n},\ldots[/ilmath]) is a subsequence

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

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

↑
Some books may simply require increasing, this is wrong. Take the theorem from Equivalent statements to compactness of a metric space which states that a metric space is compact [ilmath]\iff[/ilmath] every sequence contains a convergent subequence. If we only require that:
- [ilmath]k_n\le k_{n+1} [/ilmath]
Then we can define the sequence: [ilmath]k_n:=1[/ilmath]. This defines the subsequence [ilmath]x_1,x_1,x_1,\ldots x_1,\ldots[/ilmath] of [ilmath](x_n)_{n=1}^\infty[/ilmath] which obviously converges. This defeats the purpose of subsequences. A subsequence should preserve the "forwardness" of a sequence, that is for a sub-sequence the terms are seen in the same order they would be seen in the parent sequence, and also the "sub" part means building a sequence from it, we want to built a sequence by choosing terms, suggesting we ought not use terms twice.
The mapping definition directly supports this, as the mapping can be thought of as choosing terms