BY-MO 2018 Problem 92

For every integer $n\geqslant2$ prove the inequality $$\frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2},$$ where $k!=1\cdot2\cdot\ldots\cdot k$ .