I am having troubles dealing with following problems:
Suppose $\mathcal{F}$ is a collection of all holomorphic functions defined on a region $\Omega$ with values on the right half plane. If there exists a point $z_0 \in \Omega$ such that $f(z_0)=g(z_0)$ for all $f,g \in \mathcal{F}$, then $\mathcal{F}$ is a normal family.
Let $\mathcal{F}$ be a collection of all holomorphic functions defined on a region $\Omega$ with values in $U_0=\mathbb{C} \backslash \{x+i0, 0 \leq x \leq 1\}$. If there exists a point $z_0 \in \Omega$ such that $f(z_0)=g(z_0)$ for all $f,g \in \mathcal{F}$, then $\mathcal{F}$ is a normal family.
A hint is to use the following version of the Montel Theorem:
Let $\mathcal{F}$ be a family of holomorphic functions on $\Omega$. If every function in $\mathcal{F}$ does not assume the same two values, then $\mathcal{F}$ is a normal family.
However, I don't see any conditions in the above two problems that can imply "If every function in $\mathcal{F}$ does not assume the same two values". Any clarifications or hints will be greatly appreciated!