Def. (Subsequence)
A subsequence of $\{a_j\}_j$ in $\mathbb{R}^n$, is a new sequence, denoted $\{a_{j_k}\}_k$, where $\{j_k\}$ is an increasing sequence of integers. (Increasing means that $j_{k+1}>j_k$ for every $k$.) Thus, the $k$th term $a_{jk}$ of the subsequence is the $j_k$th term of the original sequence.
I'm trying to understand what this means, is it saying:
$\{a_{j_k}\}_{k=1}^\infty \text{ is a subsequence of }\{a_j\}_{j=1}^{\infty}\text{ iff }$
$$\color{lightblue}{(}\exists f,h:\mathbb{N}\to\mathbb{N},g:\mathbb{N}\to\mathbb{R},s.t.\forall x,j,k\in\mathbb{N},$$
$$f(x)=x\wedge g(j)=a_j\wedge h(k)=j_k$$
$$\wedge\in\mathbb{N}, h(k+1)>h(k)\color{lightblue}{)}$$
$$\rightarrow\color{lightblue}{(}\exists f,g,h:\mathbb{N}\to\mathbb{N}, s.t.$$
$$ \text{range}(g\circ f)=\{a_j\}_{j=1}^\infty$$
$$\wedge \text{range}(g\circ h\circ f)=\{a_{j_k}\}_{k=1}^\infty\color{lightblue}{)}$$
To help understanding, here i'm trying to break the notation of sequence into range of functions with domain$\backslash$codoamin $\mathbb{N}$
Is this correct, any suggestion would be appreciated.
Thanks for your help.
A sequence is a function $a : \mathbb N → \mathbb R$, i.e. $a_j=a(j)$.
In order to generate a sub-sequence we need a function $j : \mathbb N → \mathbb N$ such that $j(k+1) > j(k)$.
In conclusion :