I've seen it stated somewhere that the following theorem is a corollary of Cauchy's integral formula, perhaps also within the realm of Laurent series:
If $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic on a domain $D$ and if $z_n$ is a sequence such that $z_n\in D$ and $z_n\rightarrow z\in D$ and $f(z_n)=0$ then it follows that $f$ is identically zero.
However, I can't seem to find a proof (that is simple enough that I can read) which demonstrates the claim. I know the proof of the Liouville theorem, which seems somewhat similar and lies roughly in the same territory as this theorem, but I don't see how the proof could be adapted to this case.
Assume $f$ is not identically $0$. Let $m$ be the order of $z$ as a zero of $f$. Then $f(w)=(w-z)^mg(w)\,,\ w\in D,$ for some analytic function $g$ on $D$ that satisfies $g(z)\neq0$. In particular $g$ is continuous so there exists $\delta>0$ such that $$ |g(w)-g(z)|<\frac{|g(z)|}2 $$ for $w$ in $\mathbb{D}(z,\delta)$. Notice that this implies $g$ doesn't vanish in $\mathbb{D}(z,\delta)$. Hence $f$ doesn't vanish in $\mathbb{D}(z,\delta)\setminus\{z\}$, a contradiction since $z_n\to z$.