Holomorphic function and nth derivative.

Question

Let $K$ be a open connected subset of complex numbers and $f$ holomorphic on $K$. If $f=0$ on some open disc $D$ in $K$, then is it true that

1. $n$th derivative of $f$ is $0$ for all points in $D$

2. And this implies $f=0$ on the whole of $K$.

Progress

I know that holomorphic implies power series expansion exists about the point and was trying to see if it can be done using just this. My main question is, if $ f$ is holomorphic and zero in some disc then is it zero on any connected set containing that disc?

Answer 1

By the identity theorem both claims follow trivially.