= Solution
For fixed integers $s\leq t$, $A_s=\{\tau=s\}$ belongs to $\mathcal F_s$ and $X_s=0$ there. The <supermartingale> property and nonnegativity imply
$$
0\leq\mathbb E[\mathbf1_{A_s}X_t]\leq\mathbb E[\mathbf1_{A_s}X_s]=0.
$$
Therefore $X_t=0$ <almost surely> on $A_s$. Taking the finite union over $s\leq t$ gives $X_t=0$ on $\{\tau\leq t\}$ <almost surely>; taking the countable intersection over all integer $t$ makes the assertion simultaneous at every time. Thus \b[zero is absorbing for a nonnegative <supermartingale> in discrete time.]
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