Could you please give me some hint on how to show (if possible) that the following inequality holds?
$$\bigg|(-1)^m\frac{t^m x^m}{m!}+\sum_{j=m+1}^{\infty}\frac{(-1)^jx^jt^j}{j!}\bigg|\leq \bigg|(-1)^m\frac{t^m x^m}{m!}\bigg|$$
(where $x>0$)
notice that the left side is nothing more than
$$\bigg|e^{-tx}-\sum_{k=0}^{m-1}\frac{(-1)^jx^jt^j}{j!}\bigg|$$
Any hint or advice will be really appreciated.