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Existence of a limit $ \vec{v^*}$

\begin{eqnarray*}\Vert\vec{v}_{n+m}-\vec{v}_n\Vert & = &

Since the coefficient decreases as n increases, $\forall
> 0\ \ \exists N
> 0$ such that $\forall n \geq N, \Vert\vec{v}_{n+m}-\vec{v}_n\Vert <
Thus the series $\vec{v}_n$ has a limit.

Let us call this limit $ \vec{v^*}$ and show that $ \vec{v^*}$ is a fixed point of the operator T.

Yishay Mansour