Different proofs of $\lim_{x\to \infty}\left(1+ \frac{1}{n}\right)^n =e$

Question

I recently was teaching my friend about the number $e$. I introduced him the number by using the compound interest thing . Then I wrote down the general result -$$\lim_{x\to \infty}\left(1+ \frac{1}{n}\right)^n =e$$ The he told me that yes it works for $n=10,100,200,1000$. Beyond that his computer couldn't check . So he asked me for a formal mathematical proof of it . I thought of one but then that proof had natural logarithms - meaning they involved the number $e$ .I want to know the different ways through which this results can be proved , but without any use of $e$.

Answer 1

Here you go limit proof of e.