Definition:Extension of Sequence/Negative Integers

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Definition

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.