Limit of a geometric term

If $r<1$ and positive, and $m$ is a positive integer, shew that $(2m+1)r^m(1-r)<1-r^{2m+1}$. Hence shew that $n{r}^m$ is indefinitely small when $n$ is indefinitely great.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. V. Geometrical Progression, Examples V.b, No. 23