Series inequality $\sum _{k=n}^{\infty } \frac{1}{k!}\leq \frac{2}{n!}$

Question

Show that: $\displaystyle\sum _{k=n}^{\infty } \frac{1}{k!}\leq \frac{2}{n!}$

I am clueless here, I tried to multiply both sides with $n!$, but it doesn't make things better. I know that the left one converges against $e$ for $n=0$, but I better don't want to use its numerical value.

Answer 1

Hint: $$\frac{1}{(n+1)!} + \frac{1}{(n+2)!} + \cdots $$ $$= \frac{1}{n!} \left( \frac{1}{(n+1)} + \frac{1}{(n+1)(n+2)} + \cdots \right) $$ $$ < \frac{1}{n!} \times \text{some geometric series}$$

Answer 2

Estimate it using a geometric series.