There are two or three keys to this. And when his message and sufferings meet with spiritual life, that leads to final life. Death is unresponsive to the gospel and is confirmed in its deadness forever. Spiritual life is responsive to the gospel and is confirmed and preserved for eternal life. That is, when the revelation of the gift of righteousness meets with faith it leads to future faith.
Faith is the initial window of the soul that lets the light of the revelation of righteousness in.
And when the light of God's gift of righteousness comes in by faith, it powerfully works to awaken and sustain and engender more and more faith for the years to come. Verse 16 says that the gospel is the power of God unto salvation for all who go on believing - present tense continuous action. So we must go on believing. So if we give up on our faith and throw it away, our supposed faith will prove to have been vain, empty, dead. If that's true, now it makes clear sense why verse 17 explains how God saves believers by saying that in the gospel God reveals a righteousness for us that is first perceived and embraced by faith, and then has the effect of awakening all the necessary future faith that we need in order to be saved.