Chapter 8.1, Problem 48E

Let $U\ge 0$ be a real upper triangular $n\times n$ matrix with zeros on the diagonal. Show that
${(I_n+U)}^t\le t^n(I_n+U+U^2+\mathrm{...}+U^{n-1})$
for all positive integers t. See Exercises 46 and 47.