Subsequence/Definition

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Definition

Given a sequence $(x_n)_{n=1}^\infty$ we define a subsequence of $(x_n)^\infty_{n=1}$ ^[1]^[2] as follows:

Given any strictly increasing monotonic sequence^{[Note 1]}, (kn)∞n=1⊆N
- That means that $\forall n\in\mathbb{N}[k_n<k_{n+1}]$ ^{[Note 2]}

Then the subsequence of $(x_n)$ given by $(k_n)$ is:

(xkn)∞n=1, the sequence whose terms are: xk1,xk2,…,xkn,…
- That is to say the $i$ ^th element of $(x_{k_n})$ is the $k_i$ ^th element of $(x_n)$

Consider an (injective) mapping: $k:\mathbb{N}\rightarrow\mathbb{N}$ with the property that:

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

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Note that strictly increasing cannot be replaced by non-decreasing as the sequence could stay the same (ie a term where mi=mi+1 for example), it didn't decrease, but it didn't increase either. It must be STRICTLY increasing.

If it was simply "non-decreasing" or just "increasing" then we could define: kn:=5 for all n.
- Then $(x_{k_n})_{n\in\mathbb{N} }$ is a constant sequence where every term is $x_5$ - the 5^th term of $(x_n)$ .
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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:
- $k_n\le k_{n+1}$
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