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Condition on the information through time $n-1$. Independence of ${\xi }_n$ and the definition of the Bellman equation for terminal-wealth utility implybecause the actual holding is one candidate in the supremum. Thus $V(n,X_n)$ is a supermartingale for every strategy.

For the stated strategy it is a martingale, soEvery competing strategy has expected terminal value at most $V(0,X_0)$ by the supermartingale inequality from part (a), evaluated at the bounded time $N$. Hence ${\theta }^{\ast }$ is optimal.