I am pretty new to complex analysis and we have just studied Taylor Series, so sorry if my question is basic.
I have been trying to solve this homework problem but I am not sure my solution is formal enough.
In class we proved the following theorem:
If $f:\Omega \to \mathbb {C}$ is analytic on $\Omega$ then $f$ is representable in $\Omega$ by power series. In fact, if $ D (z_0,r) \subseteq \Omega$ (where $D(z_0,r)$ is the open disc of radius $r$ around $z_0$) then $$ f(z)=\sum\limits_{n=0}^{\infty} \frac {f^{(n)}(z_0)}{n!} (z-z_0)^n, \:\:\:\:\: z\in D(z_0,r) $$ This series is called the Taylor expansion of $f$ about $z_0$.
MY ATTEMPT:
Now, since $f(z) = e^{- \frac{1}{z^2}} $ is analytic in $\mathbb {C} \setminus \{0\}$ as a composition of analytic functions in $\mathbb {C} \setminus \{0\}$, we have from the theorem above that for a given $z_0\ne0$ our $f(z)$ is representable as a Taylor series about $z_0$. This means that the Taylor series about $z_0$ converges uniformly to $f(z)$ on each open disc $D(z_0,r)$ which does not contain $0$. This fact implies that $r$ (the convergence radius) has to be smaller than the distance between $z_0$ and $0$ which is exactly $|z_0|$. So we conclude that the radius of convergence is $|z_0|$.
END OF PROOF
Is this proof correct and formal enough?
To make it more formal I was thinking maybe I could find the Taylor expansion of $f$ about $z_0$ and then use the Cauchy-Hadamard Theorem to find the radius explicitly. To find the Taylor series I tried using the Taylor series of $g(z)=e^z$ which converges uniformly to $g(z)$ everywhere in the complex plane, and then subtitute $z$ with $-\frac {1}{z^2}$ for each $z\ne0$. To use the Cauchy-Hadamard Theorem I tried to write the series I got as $\sum\limits_{n=0}^{\infty} a_n (z-z_0)^n$ for some $a_n\in\mathbb {C}$, but then things got a bit ugly... Is all this really necessary, or one can just settle for the attempt above?
So to conclude, I would like to know if the proof above is good enough, and if not I would appreciate any idea on how to improve it or any other suggestions or hints regarding another formal proof.
Thanks in advance!