Subsequences of subsequences

real-analysissequences-and-series

A sequence $S = \{s_i\}_{i=1}^\infty$ of real numbers is a function from I (the set of positive integers) into $\mathbb{R}$ (the set of real numbers).

Definition above is given in Methods of Real Analysis by Goldberg

Say you have sequence S. You know nothing about it other than these things:
(1) It is a sequence
(2) It has subsequences

You also know those subsequences have subsequences of their own.

I am trying to prove that every subsequence of a subsequence of S is itself a subsequence of S. It seems risky to assume convergence or limit the approach to a specific set of numbers.

For example, I can say that $\{a_n\}$ is sequence in the natural numbers (every even natural number perhaps) and $\{b_n\}$ is a subsequence of $\{a_n\}$ such that $b_i$ = $a_{\mathrm{2i}}$ for all integers i. Hence $\{b_n\}$ = (4, 8, 12, 16).

Obviously, $\{b_n\}$is also a subsequence of the natural numbers, but I have sacrificed generality.

Best Answer

Let $\Bbb{N}$ be the set of natural numbers (say starting at $1$) and let $E$ be any non-empty set (for the typical case, suppose $E=\Bbb{R}$). Here are (the super formal) definitions:

  1. A sequence in $E$ is by definition a function $a:\Bbb{N}\to E$.
  2. A sequence $b:\Bbb{N}\to E$ is called a subsequence of the sequence $a:\Bbb{N}\to E$ if there exists a strictly increasing function $\theta:\Bbb{N}\to\Bbb{N}$ such that $b=a\circ \theta$ (for example if $\theta(i)=2i$, then $b(i)=a(2i)$ so the $b$ is the "even subsequence" of $a$).

So, with this definition, suppose we have three sequences $a,b,c:\Bbb{N}\to E$ and that $b$ is a subsequence of $a$, and that $c$ is a subsequence of $b$. We wish to show that $c$ is a subsequence of $a$. This is immediate, because by hypothesis, there exist strictly increasing functions $\theta_1,\theta_2:\Bbb{N}\to \Bbb{N}$ such that $c=b\circ \theta_2$ and $b=a\circ \theta_1$. Therefore, \begin{align} c&=b\circ \theta_2=(a\circ \theta_1)\circ\theta_2=a\circ(\theta_1\circ \theta_2) \end{align} Note that $\theta_1\circ\theta_2$ is a composition of strictly increasing functions $\Bbb{N}\to\Bbb{N}$, so is itself a strictly increasing function $\Bbb{N}\to\Bbb{N}$. This shows $c$ is indeed a subsequence of $a$.

Of course, standard terminology/notation demands that we write $a_n$ for the $n^{th}$ term rather than $a(n)$, and that we denote the whole sequence as $\{a_n\}_{n=1}^{\infty}$ or $(a_n)_{n=1}^{\infty}$ rather than the single letter $a$ to denote the function $\Bbb{N}\to E$.

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