How do I derive $\lim_{x\to\infty}(1+1/x)^x,\quad x\in\mathbb{R}$ from $\lim_{n\to\infty}(1+1/n)^n=e,\quad n\in\mathbb{N}$?

Question

How do I derive $$\lim_{x\to\infty}(1+1/x)^x,\quad x\in\mathbb{R}\tag{1}$$ from $$\lim_{n\to\infty}(1+1/n)^n=e,\quad n\in\mathbb{N}\tag{2}$$ ?

Note:

• Without calculus (continuity, derivative, aso.).
• (2) is only an idea, every method using limit, Cauchy, aso is good.

Answer 1

Hint: say $n \le x \le n+1$ ($x$ large). Then $$ \left ( 1 +{1\over x }\right )^x \le \left( 1 + {1\over n}\right)^{n+1}.$$ Do something similar to bound the term to limit below, and use the squeeze theorem.