Chapter 8.1, Problem 49E

Let R be a complex upper triangular $n\times n$matrix with $\left|r_{ii}\right|<1$ for $i=1,\mathrm{...}n$ . Show that

$\underset{x\to \infty }{\lim }{R}^t=0$ ,

meaning that the modulus of all entries of $R^t$ approaches zero. Hint: We can write $\left|R\right|\le \lambda (I_n+U)$, for some positive real number $\lambda <1$ and an upper triangular matrix $U\ge 0$ with zeros on the diagonal. Exercises 47 and 48 are helpful.