$ L = \lim_{ n \to \infty} n!(1-\frac{1}{e}\sum_{k=0}^{n} 1/k!) =0 \:$

I know that $ \lim_{ n \to \infty}\sum_{k=0}^{n} 1/k! =e$

So I’m assuming that $ L$ goes to $ 0$ because $ (1-\frac{1}{e}\sum_{k=0}^{n} 1/k!)$ goes to $ 0$ faster than $ n!$ goes to infinity. But how to prove this?