Faith, by definition, is not based on "proof" (because then it would not be "faith" would it?) but it is based on evidences.
There is evidence, not proof, that Jesus is the Messiah and the Creator. There is evidence, not proof, that Jesus will deconstruct this present cursed universe and will create a new eternal one in which He, the Father, and all bellievers will live together forever (Revelation 20-22).
Similarly, there is evidence, not proof, that Hell is the ongoing eternal abode of unbelievers. Nothing there ceases to exist, not even its worms die, but neither do they live in any positive sense. That hapless eternal existence is called the second death for a reason.
Is it unfair that unbelievers should spend an eternity existing in Hell? Since unbelievers hate the holy God, they would find eternity with Him intolerable. Although they will find Hell terrible, it will be more acceptable to them than spending eternity with a God whom they hate. So, is giving unbelievers a more tolerable existence really unfair, or, is it merciful?