A problem about martingale with stopping time . In Durrett's "Probability,theory and examples":
Suppose $X_n$ is supermartingale and $H_n$ is predictable. define:
$$(H\cdot X)_n\triangleq\sum^n_{m=1}H_m(X_m-X_{m-1})  $$
$N$ is stopping time and let $H_m=1_{\{N\ge m\}}$
Then the author claim:
$$(H\cdot X)_n=X_{N\wedge n}-X_0  $$
I am confused about this claim since we consider ther coefficient of $X_0$, according to the definition,the coefficient should be $-1_{\{N\ge 1\}}$,not $-1.$
If you don't understand the background you can refer to:
http://www.math.duke.edu/~rtd/PTE/PTE4_1.pdf
Page 200,the last but 3rd line.

It follows from the very definition of a stopping time (page 155) that $N \geq 1$. Hence, $$-1_{\{N \geq 1\}} = -1.$$
