In the book of The Theory of Functions by Titchmarsh, at page 88, it is first stated that
Let $f(z)$ be a function analytic in a region $D$, and let $P_1, P_2, > ...$ be a set of posts having a limit-point $P$ inside $D$. Then if $f(z)=0$ at every point $P_n$, it follows that $f(z) = 0$ at all points of $D$.
Than, as a corollary it is given the following without a proof,
If a function is analytic in a region, and vanishes at all points of any smaller region included in the given region, or along any arc of a continuous curve in the region, then it must vanish identically.
I want to show that if $f$ is analytic on $D$, and $\exists E \subseteq D$ (E is a region) s.t $f|_E=0$, then $f = 0$ on $D$.
Proof:
We can express $f$ as $$f(z) = \sum_{k=0}^\infty a_n (z-a)^n \quad \forall z \in D.$$ for some $a\in E$.
Then since we can find a sequence of point $P_i$ in $E$ that converges to $a$. Since $f(P_i) = 0$, by the previous theorem, $f = 0$ on $D$.
Note that, even $E$ was just an arc in $D$, we could still approach to a zero of $f$ with a sequence of points all of which are zero of $f$, hence by the previous theorem, $f=0$ on $D$.
I'm new to the subject, so I would really appreciate if you could point the mistakes in the proof, or tell me whether is it valid or not.