Subsequence/Definition

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Definition

Given a sequence (xn)∞n=1 we define a subsequence of (xn)∞n=1[1] as follows:

  • Given any strictly increasing sequence, (kn)∞n=1
    • That means that ∀n∈N[kn<kn+1][Note 1]

The sequence:

  • (xkn)∞n=1 (which is xk1,xk2,…xkn,…) is a subsequence

As a mapping

Consider an (injective) mapping: k:N→N with the property that:

  • ∀a,b∈N[a<b⟹k(a)<k(b)]

This defines a sequence, (kn)∞n=1 given by kn:=k(n)

  • Now (xkn)∞n=1 is a subsequence

Notes

  1. Jump up ↑ 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 ⟺ every sequence contains a convergent subequence. If we only require that:
    • kn≤kn+1
    Then we can define the sequence: kn:=1. This defines the subsequence x1,x1,x1,…x1,… of (xn)∞n=1 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

References

  1. Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin
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