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Calculate $$\lim\limits_{n\to\infty}(1-\frac{\lambda_n}n)^n$$

I came upon the calculation of $\lim\limits_{n\to\infty}(1-\frac{\lambda_n}n)^n$ when I was reviewing Poisson distribution. Notice that $\lambda_n = np_n(n\in\mathbb{Z}{\geq 0}, 0\leq p \leq 1)$ and $