Let $X_1, X_2, \ldots$ be i.i.d. r.v.s with CDF $F$, and let $M_n = \max(X_1,X_2, \ldots ,X_n)$. Find the joint distribution of $M_n$ and $M_{n+1}$, for each $n \geq 1$.
So, CDF of $M_{n+1}$ is given as, $$ \begin{align} P(M_{n+1} \leq x) &= P(X_1 ≤ x, X_2 ≤ x, \ldots, X_n ≤ x, X_{n+1} ≤ x ) \\ &= \underbrace{F(x) \times F(x) \times \ldots}_{(n+1) \text{ times}} \\ &= F(x)^{n+1} \end{align}$$
We want to consider two cases: $P(M_n \leq a,M_{n+1} \leq b)$. However, after getting these facts, I am lost. Can someone help me in finding this joint distribution? Much thanks.