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Okay so I read Richard Rusczyk's AoPS Volume 2 Book, and I stumbled upon the part where he informs very briefly that $\lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^n=e$. But he doesn't really provide a rigorous proof as to why that's true (not criticizing him or anything).. It would really help if someone could provide me with the simplest proof possible as to why $\lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^n=e$. Thank you in advance!

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