 Solution

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