Sequences and subsequences, Part II

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In this part, we will have a closer look at strictly increasing sequences of natural numbers, before beginning to discuss subsequences of sequences.

Recall that I work with the convention that $\N=\{1,2,3,\dots\}$, so that $0\notin\N$.

Let $(n_k)_{k\in\N}$ be a sequence of natural numbers (so $n_k\in\N$ for all $k\in \N$). As discussed in Part I, we may identify this sequence $(n_k)_{k\in\N}$ with a function $g:\N\to\N$, where $g(k)=n_k\;(k\in\N).$

We will be particularly interested in the case when the sequence $(n_k)$ is strictly increasing, i.e., when

$n_1<n_2<n_3<\cdots\,.$

This is equivalent to saying that the function $g:\N \to \N$ above is a strictly increasing function. This way of thinking might be helpful to us later, because of the fact that if you compose two strictly increasing functions from $\N$ to $\N$, you obtain another strictly increasing function from $\N$ to $\N$.

There is, however, another way to think about strictly increasing sequences of natural numbers. Given such a sequence $(n_k)_{k\in\N}$, consider the set $A=\{n_1,n_2,n_3,\dots\} \subseteq \N$. Note that $A$ is a set, and not a sequence. Now, in general, it is not possible to recover the original sequence from the set of terms in the sequence, because we don't know what order the sequence goes through the terms, and also there may be repeated terms in the sequence. However, the situation is different if we know that the original sequence was strictly increasing. Note that, in this case, the set of terms $A$ must be an infinite subset of $\N$.

Now let $A$ be an infinite subset of $\N$. Then (essentially because of the well ordering principle for $\N$) there is a unique strictly increasing sequence $(n_k)_{k\in\N}$ of natural numbers such that 

$A=\{n_1,n_2,n_3,\dots\}\,.$

(We just select the elements of $A$ in strictly increasing order.) So we have another one-to-one correspondence! We can work directly with strictly increasing sequences of natural numbers, or with strictly increasing functions from $\N$ to $\N$, or with infinite subsets of $\N$. We can take whichever approach makes our task easier!

We now define subsequences of sequences.

Definition of subsequence

Let $X$ be a non-empty set, and let $(x_n)_{n\in\N}$ be a sequence of elements of $X$. Then a subsequence of $(x_n)$ is a sequence of the form $x_{n_1},x_{n_2},x_{n_3},\dots$, where $(n_k)_{k\in\N}$ is a strictly increasing sequence of natural numbers.

The subsequence can also be written as $(x_{n_k})_{k\in\N}$, always remembering that we require that $n_1<n_2<n_3<\cdots$.

As a special case, consider the sequence $(x_n)_{n\in\N}$ given by $x_n=n\;(n\in\N)$. Then the subsequences of this sequence are precisely the strictly increasing sequences of natural numbers discussed above. I suppose  that this gives a fourth way to think about such sequences, as long as we don't define things circularly (we used the notion of a strictly increasing sequence of natural numbers to define what we mean by a subsequence).

In Part III we will put together everything we have discussed so far, and see how this gives us several other ways we can think about subsequences of sequences. We also discuss what it means for two subsequences to be equal: this is perhaps not completely obvious if the original sequence has repeated terms. 



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