**Proof:**

Suppose there are two nonzero *l <sup>s</sup>* >0 and *l <sup>t</sup>* >0. Taking the limit on *Rk* , *<sup>j</sup>* as *k* goes to infinity we have

$$\lim\_{k \to \infty} \text{(\(R\_{k,j}\)}=l\_j = \text{Lim}\left(\frac{c\_j R\_{k-1,j}}{\sum\_{i=1}^l c\_i R\_{k-1,i}}\right) = \frac{c\_j l\_j}{\sum\_{i=1}^l c\_i l\_i} \tag{6}$$

Now since *l <sup>s</sup>* >0 and *l <sup>t</sup>* >0, then by equation (6) we have

$$l\_s = \frac{c\_s l\_s}{\sum\_{i=1}^{l} c\_i l\_i} \quad \Rightarrow \quad c\_s = \sum\_{i=1}^{l} c\_i l\_i \text{ and } \ l\_t = \frac{c\_t l\_t}{\sum\_{i=1}^{l} c\_i l\_i} \quad \Rightarrow \quad c\_t = \sum\_{i=1}^{l} c\_i l\_i \tag{7}$$

In other words, we conclude *cs* =*ct*, which is a contradiction.
