\begin{definition}[Definition:Extension of Sequence/Negative Integers]
A sequence on $\N$ can be extended to the negative integers.
Let $\left \langle {a_k} \right \rangle_{k \mathop \in \N}$ and $\left \langle {b_k} \right \rangle_{k \mathop \in \N}$ be sequences on $\N$.
Let $a_0 = b_0$.
Let $c_k$ be defined as:
:$\forall k \in \Z: c_k = \begin{cases}
a_k & : k \ge 0 \\
b_{-k} : & k \le 0 \end{cases}$
Then $\left \langle {c_k} \right \rangle_{k \mathop \in \Z}$ '''extends''' (or '''is an extension of''') $\left \langle {a_k} \right \rangle_{k \mathop \in \N}$ to the negative integers.
Category:Definitions/Sequences
\end{definition}