$e^x \neq 0$ Proof Correctness

Question

note: I’m not looking for another proof for this, I am just asking if this proof is correct.

This is my proof.

Assume $f(x)=e^x=0$

Then $e^\frac{x}{2}e^\frac{x}{2}=0$

More generally, $(e^\frac{x}{n})^n=0$

$\lim_{n \to \infty}f(x)=\lim_{n \to \infty}1^n=1$

That means $1=0$ hence the contradiction.

Answer 1

The proof is not correct because it assumes implicitely that if a sequence $(a_n)_{n\in\mathbb N}$ converges to $1$, then $\lim_{n\to\infty}{a_n}^n=1$. This is not true. For instance,$$\lim_{n\to\infty} \left (1 + \frac1n \right )=1\text{ and }\lim_{n\to\infty}\left(1 + \frac1n\right)^n=e\neq1.$$