Limit in Big-O in the proof of Poisson limit theorem

Question

while I was studying the proof of the Poisson limit theorem (One can find it here), I've got confused with one thing: $$ \lim_{n \to \infty} \frac{n^k + O\left(n^{k-1}\right)}{k!}\frac{\lambda^k}{n^k}\left(1-\frac{\lambda}{n}\right)^{n-k} = \lim_{n\to\infty} \frac{\lambda^k}{n^k}\left(1-\frac{\lambda}{n}\right)^{n-k}. $$ How to show that $$ \lim_{n \to \infty} \frac{n^k + O\left(n^{k-1}\right)}{k!} = 1? $$ Thanks in advance!

Answer 1

You've made an error. It should be

$$ \lim_{n \to \infty} \frac{n^k + O\left(n^{k-1}\right)}{k!}\frac{\lambda^k}{n^k}\left(1-\frac{\lambda}{n}\right)^{n-k} = \lim_{n\to\infty} \frac{\lambda^k}{k!}\left(1-\frac{\lambda}{n}\right)^{n-k}. $$

Which is not difficult.