I have a question that may seem trivial.
I am studying Kelley's General Topology. I was reading Chapter 2 when I came across an unusual definition. Tipically, I define a subsequence as follows:
a subsequence of the sequence $(a_n)_{n \in \mathbf{N}}$ is any sequence of the form $(a_{n_k})_{k \in \mathbf{N}}$ where $(n_k)_{k > \in \mathbf{N}}$ is a strictly increasing sequence of positive integers.
Whereas in the book, it says that:
$T$ is a subsequence of a sequence $S$ iff there is a sequence $N$ of non-negative integers such that $T = S \circ N$ (equivalently, $T_i = > S_{N_i}$ for each $i$) and for each integer $m$ there is an integer $n$ such that $N_i \geq m$ whenever $i \geq n$.
I can deduce that this two definitions are not the same and it may seem that one definition implies the other. Am I right or wrong? Can someone formally explain me the difference and how these two are related?
Thank you :)
The first definition in your question is the usual one. Kelley's definition would agree with it if $N$ were required to be a strictly increasing sequence, but Kelley's requirement is weaker than that. So any subsequence in the usual sense is also a subsequence in Kelley's sense, but the converse is not true.
I think that Kelley uses his unusual definition in order to agree with the treatment of nets and subnets (a sort of generalized sequence, important in general topology) later in the book. The definition of "subnet" is the same as Kelley's definition of "subsequence" with the index set $\mathbb N$ for sequences replaced by more general ("directed") partially ordered sets.