Convergence of probabilistic process

Question

Take a random vector $V$ of length $n$ where $V_i \in [m]$. Define a random process as follows. Repeatedly pick two indices $i, j \in [n]$ uniformly at random and let $V_i := V_j$ (that is change the value at index $i$ to be the value at index $j$). How can you prove that this eventually converges to a vector with all entries the same with probability $1$?

Answer 1

The quickest way is to notice that if the sequence of pairs $(2,1), (3,1), \dots, (n,1)$ ever occurs then $V_1$ will be copied to all entries in the vector.

As the choices of pairs are iid the sequence will occur eventually with probability one.