A transfinite sequence is a function whose domain is an ordinal.
If $s : \alpha \to X$ is transfinite sequence for some ordinal $\alpha$:
$s$ an $\alpha$-sequence.
$\alpha$ is the length of $s$.
$s$ is a transfinite sequence in $X$.
$s$ is also denoted by
\[\langle s_\xi : \xi < \alpha \rangle \quad\text{or}\quad \langle s_\xi \rangle_{\xi < \alpha},\]where $s_\xi$ stands for $s(\xi)$.
A countably infinite sequence is an $\omega$-sequence.
A countably infinite sequence $s$ is also denoted by
\[\langle s_n : n \in \N \rangle \quad\text{or}\quad \langle s_n \rangle_{n \in \N}.\]
A finite sequence is an $n$-sequence for some natural number $n$.
If $s$ is an $\alpha$-sequence for some ordinal $\alpha$, the extension of $s$ by $x$ is
\[s^\frown x = s \cup \{(\alpha,x)\}.\]