Solution
= Solution
If $G\geq0$ were an arbitrage gain, then $X_1^*+tG$ would be attainable from the same initial wealth for every $t>0$. Since $U$ is increasing,
$$
U(X_1^*+tG)\geq U(X_1^*)
$$
almost surely, with strict inequality on a set of positive probability under the usual strictly increasing meaning of utility. This contradicts optimality of $X_1^*$. Hence existence of an optimizer implies <arbitrage>[no-arbitrage].