Explaining maths

 Recall from Part II 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 $n_1<n_2<n_3<\cdots$. In this setting, as discussed last time, we may apply the definition above to the sequence $(n_k)_{k\in\N}$, and we see that $(n_k)_{k\in\N}$ is a subsequence of the sequence of  $1,2,3,\dots$. Now let us revisit this in terms of functions from $\N$ to $X$ and functions from $\N$ to $\N$. We can define $f:\N\to X$ by $f(n)=x_n$, and we can define $g:\N\to\N$ by $g(k)=n_k$. Then $g$ is a strictly increasing  function from $\N$ to $\N$, and we have  $x_{n_k}= x_{g(k)}=f(g(k))\;(k\in\N)$. Thus, if we define $h:\N\to X$ by $h(k)=x_{n_k}$, we have...

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 num...

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...