- #1

binbagsss

- 1,278

- 11

## Homework Statement

Hi

I am looking at this proof that , if on an open connected set, U,there exists a convergent sequence of on this open set, and f(z_n) is zero for any such n, for a holomorphic function, then f(z) is identically zero everywhere.

##f: u \to C##Please see attachment

## Homework Equations

## The Attempt at a Solution

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I follow the proof on the open disc, but am just stuck on the ' by using connectedness we can extend this to ##U## .'

I have never taken a class in topology, nor, don't think I've ever had the definition of connectedness occur in a class. But from a quick google I see it is a set that cannot be partitioned into two nonempty subsets which are open in the relative topologyinduced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.

I'm unsure how to,apply this to see that connnecteness implies that if the proof holds on an open disc, it,holds on ##U##

Many thanks