 Solution

The strategy $\eta$ is a self-financing portfolio because ${\xi }_t^{\eta }=0$, and its price is known one period in advance. Suppose its price first became nonpositive. On the event, known immediately before that date, that the next price is nonpositive while the current price is positive, an investor can short or buy the self-financing portfolio with the sign that gives no downside, finance the position at the current date, and close it at the known next price. This gives a nonnegative gain and a strictly positive gain whenever the price changes sign or reaches zero from a positive value.

More formally, stopping and scaling $\eta$ on the first such predictable event constructs an arbitrage. Since the market has no arbitrage and ${\pi }_0^{\eta }>0$, induction over dates gives