However, for this cause I obtained mercy, that in me first, Jesus Christ might display all his patience, for an example of those who were going to believe in him for eternal life.
Believe on him unto eternal life ... This strongly suggests  Romans 10:10,11; and significantly "believing on" Christ in both passages is "unto" eternal life, and salvation, as is ever the case in the New Testament.  The sacred writers were diligent never to leave an impression that merely "believing on" the Lord Jesus Christ surely led to eternal life, but merely in the direction of it, "unto life."  The apostle John gave the classical example of a case in which it did not bestow eternal life (John 12:42,43); but in even that instance "believing on" the Lord led in the direction of it.