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''$k$-tuple sequence, of elements of a given set $X$''

A mapping of the $k$-th power $\mathbb{N}^k$ of the set of natural numbers $\mathbb{N}$ into the set $X$.  An element (or term) of a multiple sequence $f : \mathbb{N}^k \rightarrow X$ is an ordered set of $k+1$ elements $(n_1,\ldots,n_k,x)$, where $x = f(n_1,\ldots,n_k)$, $x \in X$, $(n_1,\ldots,n_k) \in \mathbb{N}^k$, i.e. $n_j \in \mathbb{N}$ ($j = 1,\ldots,k$). The element is also denoted by $x_{n_1 \ldots n_k}$.