Sequences and subsequences, Part I

Sequences and subsequences play an important role throughout mathematics, and especially in mathematical analysis. In this series (or sequence?) of posts, I will combine some standard theory with some of my own musings on the subject.

These posts will be mainly about the theory around sequences etc. But to understand mathematics properly, you also need to play around with lots of examples and look at their properties. Here let me plug some old videos on Core Topics in University Mathematics (YouTube PlayList) that a group of us from Nottingham put together (funded by a teaching grant of Steve Cox). Among other videos there you can find two videos by Dan Nicks on Convergence of sequences, and three videos of my own on Sequences and their properties (including discussion of a few examples, and a quiz). I also have a video there on the topic Think of a function.

For the purposes of these posts, note that I work with the convention that $\N=\{1,2,3,\dots\}$, so that $0\notin\N$.

Let $X$ be a (non-empty) set. By default I will be working with sequences of the form $(x_n)_{n\in\N}$ where the $x_n$ are elements of $X$. Such a sequence may also be denoted by $(x_n)_{n=1}^\infty$ or $x_1,x_2,x_3,\dots$, or even $(x_n)$ for short (with the default assumption that $n$ runs through $\N$). By abuse of notation we may sometimes write $(x_n)\subseteq X$ for a sequence of elements of $X$, but the sequence $(x_n)_{n\in\N}$ is really not the same thing as the set of values $\{x_n:n\in\N\}.$

We say that two sequences $(x_n)_{n\in\N}$ and $(y_n)_{n\in\N}$ of elements of $X$ are equal if, for all $n\in\N$, we have $x_n=y_n$. 

Another way of thinking about a sequence of elements of $X$ (which could also be used as part of a formal definition) is as a function from $\N$ to $X$. Indeed there is a bijection (one-to-one correspondence) between functions $f:\N\to X$ and sequences $(x_n)_{n\in\N}$ of elements of $X$, as follows.

Given a function $f:\N\to X$, we can define a corresponding sequence $(x_n)_{n\in\N}$ of elements of $X$ by setting $x_n=f(n)\;(n\in\N)$. In the other direction, given a sequence $(x_n)_{n\in\N}$ of elements of $X$, we can define a corresponding function $f:\N\to X$ by setting $f(n)=x_n\;(n\in\N)$. 

There is a  subtle issue here, which you might not be worried about.

<Begin optional discussion of subtle issue>

Suppose that $X$ and $Y$ are non-empty sets, and that $X$ is a proper subset of $Y$. Let $f$ be a function from $\N$ to $X$. Since $X\subseteq Y$, we can define a function $g$ from $\N$ to $Y$ by $g(n)=f(n) \;(n\in\N)$. We can then set $x_n=f(n)=g(n)\in X\subseteq Y\;(n\in\N)$ to give us a sequence $(x_n)_{n\in\N}$ which is a sequence of elements of $X$, but which is also a sequence of elements of $Y$. However, the codomains of $f$ and $g$ are different, so in theory (depending on the definition of function you are working with), $f$ and $g$ are not the same function. These two functions do, however, have the same graph:

$\{(n,f(n)):n\in\N\} = \{(n,g(n)):n\in\N\}$

and this is enough for us to say that the sequences corresponding to $f$ and $g$ are equal (the sequence $(x_n)_{n\in\N}$ above). So, if you want a slightly safer formal definition of a sequence, it might be better to use graphs of functions rather than the functions themselves, if you include the codomain as part of your definition of function. 

 

<End optional discussion of subtle issue>

We are free to use different letters for our sets and for our indices. So, for example, we might have a sequence $(a_k)_{k\in\N}$ of elements of a set $B$. Indeed, in the posts which follow, we will be particularly interested in strictly increasing sequences of natural numbers. These are sequences of positive integers $(n_k)_{k\in\N}$ such that $n_1<n_2<n_3<\cdots$. This type of sequence will be relevant when we define subsequences in Part II.