In Real Mathematical Analysis by Charles Pugh, sequences are denoted using the standard notation $(a_n)$, which represents $(a_1, a_2, \dots)$. Furthermore, sub-sequences are denoted by $(a_{n_k})$ which expands to $(a_{n_1}, a_{n_2},\dots)$ i.e. $n_k \in \mathbb{N}$ indicates the index (from the mother sequence) of the $k$-th term of the sub-sequence. These two notations are thus far standard.
Finally, the confusion arises with the notation of sub-sub-sequences, which are denoted by $(a_{n_{k(l)}})$. The expansion seems to be on the $l$ variable, so that $(a_{n_{k(l)}}) = (a_{n_{k(1)}}, a_{n_{k(2)}}, \dots)$. If this expansion is correct, what is $n_{k(1)}$ supposed to indicate? Is $k$ now a function, where $n_{k(i)} \in \mathbb{N}$ denotes the index of the $i$-th term in the sub-sub-sequence (from the mother sequence, or grandmother sequence)? Is this notation standard?
Best Answer
The values $n_1,n_2,...$ represent the values of a strictly increasing $f:N\to N,$ with $n_j=f(j).$ The values $k_1,k_2,...$ also represent the values of a strictly increasing $k:N\to N.$
So $a_{n_{k(j)}}=a_{f(k(j))}.$