how to express finite set of finite sequences

Question

I want to define a finite set of finite sequences and they may have distinct cardinality. Is it correct to express this as follows:

Let $S=\{x_{n_{i}}: n=0,1,...,m \quad \text{and} \quad i=0,1,...,n_i\}$

I mean for a fixed $n$ we have a sequence of $i$ elements.

Answer 1

Normally "double subscript" notation $n_{i}$ means you have a function $n$ of an index $i$, and $n_{i} = n(i)$, which seems not to be what you require.

It sounds as if you want double indices, $$ S = \{x_{n, i}: i = 0, 1, \dots, i_{n},\ n = 0, 1, \dots, m\}. $$ In longhand: \begin{align*} &x_{0,0}, x_{0,1}, \dots, x_{0,i_{0}}; \\ &x_{1,0}, x_{1,1}, \dots, x_{1,i_{1}}; \\ &\qquad \vdots \\ &x_{m,0}, x_{m,1}, \dots, x_{m,i_{m}}. \end{align*}