Before revision:  As application of our version of the FTUM, we establish equivalence between the No-Unbounded-Profit-with-Bounded-Risk condition  , the existence of  num\'eraire portfolio, and the existence of solution to the utility maximization under equivalent probability measure  .
After revision:  As application of our version of the FTUM, we establish equivalence between the No-Unbounded-Profit-with-Bounded-Risk condition , the existence of num\'eraire portfolio, and the existence of solution to the utility maximization under equivalent probability measure for any "nice" utility and positive initial capital. Herein, a 'nice" utility is any smooth von URLenstern utility satisfying Inada's conditions and the elasticity assumptions of Kramkov and Schachermayer .