Jordan theorem : the boundary statement

Question

The Jordan theorem states that if $\Gamma$ is a closed Jordan curve in $C$, then the complement $C\setminus \Gamma$

1. has two connected components
2. one is bounded, the other is unbounded
3. the boundaries of these two components is $\Gamma$.

I found a proof of points 1 and 2.

Now I'm searching for point 3.

The two connected components of $C\setminus \Gamma$ are open, so that their boundaries are contained in $\Gamma$.

QUESTION : how to prove the reverse inclusion ? I want to prove that if $A$ is one of the two connected components of $\mathbb{C}\setminus \Gamma$, then every point of $\Gamma$ is in the boundary of $A$.