If $j \leq k$ and $\tau$ a stopping time, then $E(X_{j}1_{\tau = j})\leq E(X_{k}1_{\tau = j})$

Question

I am attempting to prove:

If $j \leq k$ and $\tau$ a stopping time, then $E(X_{j}1_{\tau = j})\leq E(X_{k}1_{\tau = j})$

I do not know how I can deduce this without any other assumptions on the process $(X_{n})_{n \in \mathbb N}$

On the assumption that $(X_{n})$ is a submartingale:

$E(X_{k}1_{\tau = j})=E(E(X_{k}1_{\tau = j}\vert \mathcal{F}_{j}))=E(1_{\tau = j}E(X_{k}\vert \mathcal{F}_{j}))\geq E(X_{j}1_{\tau = j})$

Answer 1

Certainly false without any assumption on $(X_n)$. Take $j=1,k=1, \tau =1$ and $X_1=1,X_2=0$.