I tried to prove that if $R$ is transitive then its inverse is transitive as well. $\begin{align} & a{{R}^{-1}}b\,\,,\,\,b{{R}^{-1}}c \\ & \Rightarrow bRa\,\,,\,\,cRb \\ & \Rightarrow cRa \\ & \Rightarrow a{{R}^{-1}}c \\ \end{align}$
is this a correct proof (or am I completely wrong and it's not even true?)