Positive submartingales Let $\{X_n\}$, $n>0$
be a positive submartingale with $X_{0} = 0.$ Let $V_n$ be random variables
such that 


*

*$V_n \in\mathcal F_{n−1}$ for all $n \geq 1$.

*$B > V_1 > V_2 > \dots > 0$ for some constant $B$.
Prove that for $λ > 0$
$P(\max V_{j}X_{j} > λ)$ $\leq$ $λ^{-1}$ $\sum_{j=1}^{n}$ $E(V_{j} (X_{j} − X_{j−1}))$.


I know that I need to show the above assumptions imply $\{\max V_{j}X_{j} > λ\} ⊆ \{\max Y_{j} > λ\}$ where $Y_{j}$ = $\sum_{i=1}^{j}$ $V_{i}(X_{i} − X_{i−1})$  but not sure how to show this or how it proves result. 
 A: Define $A_i:=\{ V_iX_i\gt \lambda\}\cap \{\max_{ l\lt i} V_lX_l\leqslant \lambda \} .$
The sets $(A_i)_{i=1}^n$ are pairwise disjoint and their union is the set $\{ \max_{s\leqslant n}V_nX_n\gt \lambda \} $. Since $\lambda \mathbf 1(A_i)\leqslant \mathbb E[\mathbf 1(A_i)V_iX_i]$, it follows that 
$$\tag{1 }\mathbb P\left(\max_{s\leqslant n}V_nX_n\gt \lambda\right) \leqslant \frac 1{\lambda}\sum_{i=1}^n    \mathbb E[\mathbf 1(A_i)V_iX_i].
$$
Define $B_j :=\bigcup_{i\leqslant j}A_i$; then \begin{align}
\sum_{i=1}^n    \mathbb E[\mathbf 1(A_i)V_iX_i]&=\sum_{i=1}^n    \mathbb E[\mathbf 1(B_i)V_iX_i]-\mathbb E[\mathbf 1(B_{i-1} )V_iX_i]\\
&=\sum_{j=1}^n    \mathbb E[\mathbf 1(B_j)V_jX_j]-\sum_{j=0}^{n-1}     \mathbb E[\mathbf 1(B_j)V_{j+1}  X_{j+1 } ]\\ 
\tag{2 }&=\mathbb E[\mathbf 1(B_n)V_nX_n]+\sum_{j=1}^{n-1}     \mathbb E\left[\mathbf 1(B_j)\left(V_jX_j-V_{j+1}X_{j+1}    \right)\right].
\end{align}
Now, we write 
$$\mathbb E\left[\mathbf 1(B_j)\left(V_jX_j-V_{j+1}X_{j+1}    \right)\right]=
\mathbb E\left[\mathbf 1(B_j)\left(V_j-V_{j+1} \right)  X_j   \right]
+\mathbb E\left[\mathbf 1(B_j)V_{j+1}\left(X_j-X_{j+1}    \right)\right].$$
Exploiting the fact that $B_j$ belongs to $\mathcal F_j$ and the submartingale property, we observe that the second term is non-negative.
Since the sequence $(V_j)_{j=1}^n$ is decreasing, we derive the bound 
$$\mathbb E\left[\mathbf 1(B_j)\left(V_jX_j-V_{j+1}X_{j+1}    \right)\right]\leqslant \mathbb E[(V_j-V_{j+1})X_j],$$
hence by (1) and (2), 
$$\mathbb P\left(\max_{s\leqslant n}V_nX_n\gt \lambda\right) \leqslant \frac 1{\lambda}\left(\sum_{j=1}^{n-1} \mathbb E[(V_j-V_{j+1})X_j]+\mathbb E[V_nX_n]\right),$$
which is the wanted result up to an Abel's transform.
