To be found: $$\lim \left(1+\frac{2}{n}\right)^n$$

Presuppose $~~\lim \left(1+\frac{1}{n}\right)^n=e~~$ is already shown.

Expanding the first equation: $$\lim \left(1+\frac{1}{\frac{n}{2}}\right)^{2\cdot\frac{n}{2}}=\lim \left(1+\frac{1}{\frac{n}{2}}\right)^{\frac{n}{2}}\cdot\lim \left(1+\frac{1}{\frac{n}{2}}\right)^{\frac{n}{2}}$$

I'd say that those two factors converge against $e$, just not as fast. My explanation would be, that the sets $\mathbb N$ and $\{\frac{n}{2}:n\in \mathbb N\}$ have infinitely many elements in common.

So is it enough to just (state that and) add
$$=\lim \left(1+\frac{1}{n}\right)^n\cdot \lim \left(1+\frac{1}{n}\right)^n=e\cdot e=e^2$$?