View source for A sequence consisting of the nth terms of the sequences in a Cauchy sequence of elements in any little-L space is itself a Cauchy sequence of complex numbers

{{DISPLAYTITLE:A sequence consisting of the {{M|n^\text{th} }} terms of the sequences in a Cauchy sequence of elements in any {{M|\ell^p}} space is itself a Cauchy sequence of complex numbers}}
: This theorem is a [[precursor]] to [[the little-L spaces are complete]]
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==Statement==
Let {{M|p\in[1,+\infty]\subseteq\overline{\mathbb{R} } }} be given and consider the [[little-L space|{{M|\ell^p}} space]] accordingly. So elements in {{M|\ell^p}} are {{M|(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C} }} such that certain properties hold. 

Let {{M|\big((x_n^k)_{n\in\mathbb{N} }\big)_{k\in\mathbb{N} } \subseteq\ell^p}} be a [[Cauchy sequence]]
* {{ie}} {{M|\big((x^1_n)_{n\in\mathbb{N} },(x^2_n)_{n\in\mathbb{N} },\ldots,(x^k_n)_{n\in\mathbb{N} },\ldots\big)\subseteq\ell^p}} is a Cauchy sequence
Then we claim{{rFAVIDMH}}:
* for all {{M|k\in\mathbb{N} }} that {{M|x:\eq(x^n_k)_{n\in\mathbb{N} }\subseteq\mathbb{C} }} is itself a Cauchy sequence
** Notice that {{M|x:\eq(x^n_k)_{n\in\mathbb{N} } :\eq (x^1_k,x^2_k,\ldots,x^n_k,\ldots)}} - the {{M|k^\text{th} }} term of each element in {{M|x}}
==Proof==
{{Requires proof|grade=E|msg=I've just done it on paper, upload that!
<gallery>
File:CauchySequenceOfNthsTermsOfCauchySequenceInEllpProof.jpg|Proof on paper!
</gallery>
}}
==References==
<references/>
{{Theorem Of|Functional Analysis|Analysis}}