How to prove $\lim_{n\to\infty}(1+1/n)^n=\lim_{n\to\infty}1+1/1!+1/2!+...+1/n!$ [closed]

How to prove $\lim_{n\to\infty}(1+1/n)^n=\lim_{n\to\infty}1+1/1!+1/2!+...+1/n!$ rigorously?

I have read many threads regarding the natural constant e, but couldn't find out how to prove this.