Problem 3.24 of "Brownian Motion & Stochastic Processes" by Karatzas and Shreve - Submartingales and stopping times

I think you have nearly answered your own question. The finite nature of the inequality in (ii) follows from the finite property of the expectation of the supremum, i.e. $$ \int\limits_{ \{ 0\leq T \leq t \} } |X_T| \ dP \leq E\left[ \sup_{ 0\leq u \leq t } |X_u|\right] < \infty, $$ and hence, $$ E\left[ |X_{T\wedge t}| \right] \leq E\left[ \sup_{ 0\leq u \leq t } |X_u|\right] + E[|X_t|] < \infty. $$ To get condition (iii), apply the conditional expectation value to both sides of the inequality. This gives, $$ E\left[ X_{T\wedge t} | F_S \right] \geq E\left[X_{T\wedge S} | F_S \right], $$ which follows by definition and the given fact that $S\leq T$.