sequence

Sequences

Given any set $X$, a sequence in $X$ is a function $f:\mathbb{N}\to X$ from the set of natural numbers to $X$. Sequences are usually written with subscript notation: $x_0,x_1,x_2\mathrm{\dots }$, instead of $f(0),f(1),f(2)\mathrm{\dots }$.

Generalized sequences

One can generalize the above definition to any arbitrary ordinal. For any set $X$, a generalized sequence or transfinite sequence in $X$ is a function $f:\omega \to X$ where $\omega$ is any ordinal number. If $\omega$ is a finite ordinal, then we say the sequence is a finite sequence.