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

$$\begin{eqnarray*}\Vert\vec{v}_{n+m}-\vec{v}_n\Vert & = & \Vert\sum_{k=0}^{n-1}\... ...\lambda^n(1-\lambda^m)}{1-\lambda}\Vert\vec{v}_1-\vec{v}_0\Vert \end{eqnarray*}$$

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