Here is a context free grammar that I have been given for practice: Grammar $G = (V,\Sigma,R,S)$ where $V$ is $\{S,A,B,a,b,c\}$ and $\Sigma$ is $\{a,b,c\}$. $R$ has the following rules:
$$\begin{align}S &\to A\\ A &\to BB\\ S &\to bS\\ S &\to cS\\ A &\to a\end{align}$$
I am fairly sure that it is not ambiguous, since for a grammar to be ambiguous there will be two ways to reach a result. In the grammar above, each $b$ comes from $S \to bS$, each $c$ comes from $S \to cS$, and each $a$ comes from $S \to A \to a$. It seems the $A \to BB$ is never used and $c=(b\cup c)^*a$. So there is only one derivative of each string in $L$.
My question is, is my thinking correct and would this suffice to prove that the grammar is not ambiguous?
Edit: We went over the problem in class a while back and the explanation above was accurate.