Let be $f, g: \mathbb{C} \to \mathbb{C}$ analytic functions.
If $f(\frac{1}{n}) = 0$ $\forall$ $n \, \in \, \mathbb{N}$ then $f(z) = 0$ $\forall$ $z \, \in \, \mathbb{C}$.
If $g(z) = 0$ $\forall$ $z$ in a subset of $\mathbb{C}$ which has a accumulation point then $g(z) = 0$ for all complex numbers.
I can see that first assumption is from second, but why are they true ?
This is a consequence of the famous Identity theorem (https://en.wikipedia.org/wiki/Identity_theorem), one of the most powerful tools in complex analysis!