Let $R$ is a binary relation on set a $S$ and let $R_0, R_1, R_2 \dots$ be defined as follows.
$$\begin{align} R_0 & := \Delta_{S} = \{(x,x) : x ∈ S\} \\ R_{n+1} & := R_n ∪ (R;R_n), && \text{for } n \geq 0 \end{align}.$$
There exists $i ∈ N$ such that $R_i = R_{i+1}.$
If $|S| = k,$ show that $(R ∪ R^{-1}) _{k^2}$ is an equivalence relation.
I know that I just have to prove that $(R ∪ R^{-1}) _{k^2}$ is reflexive, symmetric and transitive. But with the given information, I don't know how to approach this to come up with a formal proof.