All believe that He lives and is eternal, He is good and does good to the wicked and to the good -- Rosh Hashanah Musaf Prayer.
How does His doing good to the wicked prove that he is eternal?
A king of flesh and blood who is eventually going to die rushes to take vengeance from his enemies during his lifetime, for either he or his enemy might die, and he would miss the chance to be avenged. Since G-d lives forever He does not have to rush to take vengeance, because He can collect whenever He wants, even after the person’s death, if necessary.1
G-d does not desire that the wicked perish, but rather that they repent and live. Thus, He patiently does good even to the wicked, though they do not deserve it, and does not rush to take His vengeance. And as for the punishments they deserve -- if they do not repent? G-d is eternal, and never "misses" the chance to make an individual pay for his deeds.2
The words "to the good" are superfluous; if He does good to the wicked, He surely does good to the good too?
Answer:
Some people are aware that they were wicked during the year and resolve on Rosh Hashanah to repent and change their ways in the future. There are, however, others who don’t even realize the wickedness of their ways, thinking that they are good and what they are doing is good.
G-d has mercy on every Jew. He is good and does good not only to the wicked -- who are cognizant of their past and resolve to improve, but also to those [who in their own eyes] are good -- and who see no reason to change or improve.3
