Conditions for a family to be normal What are the conditions that must satisfy $A$ and $B$ (with $A,B\subset\mathbb{C}$) such that the family 
$$\mathfrak{F}_{A,B} = \lbrace f(z)=az+b : a\in A, b\in B\rbrace , \ z\in\mathbb{C}$$
to be normal in $\mathcal{H}(\mathbb{C})$, where $f(z)$ is a entire function.
My idea is to apply Montel's theorem: for every $K\subset\mathbb{C}$ there is $M_{K}$ such that $|f(z)|\leq M_{K}$ for every $z\in K$. Then I obtain that the conditions are that $A$ and $B$ have to be bounded for which the family is normal.
This argument is correct?, or can you give me a hint of how to solve this of a more explicit way? 

As Martin R pointed out, your argument is hard to judge without details. After adopting Montel's theorem, having deduced that it is necessary for $\mathfrak F_{A,B}$ to be normal that for each compact $K\subset\mathbb C$, $|f(z)|\leqslant M_K$ for some positive $M_K$ whenever $f\in\mathfrak F_{A,B}$ and $z\in K$, you can pick a compact $K_0$ that contains $0$ and $1$. Then consider $|f(0)|$ and $|f(1)|$ for each $f\in\mathfrak F_{A,B}$ and this gives the boundedness of $A$ and $B$.
