I have some queries regarding the proof of maximum modulus principle in complex analysis.I want know whether my proof is correct.
Proof:
Let if possible,$|f(z)|$ attain maximum at $z=z_0$ i.e. $|f(z)|\leq |f(z_0)|$ for all $z\in \Omega$,where $\Omega$ is the domain of $f$ which is open connected.Then $z_0$ is contained within a disc $D_{z_0}$ contained in $\Omega$,by open mapping theorem,then image $f(D_{z_0})$ is open in $\mathbb C$.Now consider $f(z_0)$ ,take a disc around $f(z_0)$ such that it is contained in $f(D_{z_0})$.Now joint the origin with $f(z_0)$ ,the length of the line segment gives $|f(z_0)|$ now extend this line so that it meets the boundary of the open disc centred at $f(z_0)$.The point on the boundary of that disc corresponds to some $f(z_1)$ such that $|f(z_1)|>|f(z_0)|$ which is a contradiction.
Is this proof alright?