Sequences and subsequences, Part IV
Recall from part III that we are using the following notation, which may not be entirely standard: Let $X$ be a non-empty set, and let $(x_n)_{n\in \N}$ be a sequence of elements of $X$. Given positive integers $n_1<n_2<n_3<\cdots$, define the infinite set $A \subseteq \N$ by $A=\{n_1,n_2,n_3,\dots\}$. Then $(x_n)_{n\in\A}$ denotes the subsequence $x_{n_1}, x_{n_2}, x_{n_3},\dots$ of $(x_n)$. Note (again) that we are working with a one to one correspondence between infinite subsets of $\N$ and strictly increasing sequences of natural numbers here, and so every subsequence of $(x_n)$ does have this form. (However, with our conventions, if the sequence $(x_n)$ has repeats, then different subsets of $\N$ might give the same subsequence.) Note also that if $A=\N$ then (to our relief) the notation $(x_n)_{n\in A}$ is unambiguously equal to the original sequence $(x_n)_{n\in \N}$. Now let's look at convergence of sequences of real numbers and connections with subsets of $\N$. (A...