Notational quibble regarding the indexes of subsequences

Question

When we want to define a subsequence of a $\{a_n\}_{n\in \mathbb{N}}$, i.e. $\{b_{n_k}\}_{k\in\mathbb{N}}$, we're relying on a function $k: \mathbb{N} \to \mathbb{N}$ (let's say $k(n) = 2n-1$), such that $b(k) = a \circ k(n)$. But then how do we refer to $\{b_{n_k}\}$'s terms? For instance, which is the term that corresponds to $a_3$? Is it $b_{n_2}, b_{1_k}$, etc.? Which subindex is fixed and which isn't?

Answer 1

The sequence $\{a_n\}_{n \in \Bbb N}$ is a map $a\colon \Bbb N \to \Bbb R$ that takes $n \in \Bbb N$ to $a_n = a(n) \in \Bbb R$. If you want write $\{b_{n_k}\}_{k \in \Bbb N}$, it is understood that we have a composition $k \mapsto n(k) = n_k \mapsto b(n_k) = b_{n_k}$. That is to say, you have an increasing function $n\colon \Bbb N \to \Bbb N$ that takes $k$ to $n(k) = n_k$ and a sequence $b:\Bbb N \to \Bbb R$. Then the relevant composition is $b\circ n$. Do note the abuse of notation: in the first moment $n$ is an element in $\Bbb N$, while later $n$ is a function. We let $k$ range over $\Bbb N$, not $n$.