Prove that there is no non-constant analytic function $f$ on $\mathbb{C}$ such that $f(z)=0$ for all $z \in \mathbb{R}$.
Doesn't this follow immediately by the identity theorem? There is a sequence along the real axis that converges to 0...
Attempt
Suppose $f$ is entire and $f(z)=0$ for all real numbers. Now let $E=\mathbb{D}$. Since the set of points of $f=0$ has a limit point in $E$ we conclude $f$ is identically 0.
Indeed “this follow[s] immediately by the identity theorem”. Just take, for instance $(\forall n\in\mathbb{N}):a_n=\frac1n$. Then $\lim_{n\in\mathbb N}=0$ and $(\forall n\in\mathbb{N}):f(a_n)=0$. Therefore, by the identity theorem, $f\equiv0$.