Eternal existence does exist in the lake of fire.
Which has nothing to do with the OP. The issue is clear: can eternal LIFE (not existence) exist in the LoF, also called the second DEATH.
The answer is absolutely NO. It cannot. Yes, there will be people in the LoF, and they will exist forever. But if one reads the OP carefully, they will understand the meaning of "death", which is separation from God. They will be separated from God for eternity.
God IS eternal life (1 Jn 5:20). Those who possess eternal life from God will live with God for eternity. Those who never possessed eternal life will be separated from God for eternity. That is called the "second death" by Scripture.
The point of the OP is that eternal LIFE cannot DIE.
If one disagrees, one must show from Scripture that such life (eternal) can or does die.
If any Christian receives the mark of the beast, they shall be tormented with fire and brimstone in the presence of the holy angels and in the presence of the Lamb, and the smoke of their torment ascends forever and ever; and they have no rest day or night, who worship the beast and his image, and whoever receives the mark of his name. Here is the patience of the saints...
Does the verse (you failed to cite your evidence) speak of Christians/believers, or just anyone who receives the mark fo the beast?
9 Then a third angel followed them, saying with a loud voice, "If anyone worships the beast and his image, and receives his mark on his forehead or on his hand, 10 he himself shall also drink of the wine of the wrath of God, which is poured out full strength into the cup of His indignation. He shall be tormented with fire and brimstone in the presence of the holy angels and in the presence of the Lamb. 11 And the smoke of their torment ascends forever and ever; and they have no rest day or night, who worship the beast and his image, and whoever receives the mark of his name." 12 Here is the patience of the saints; here are those who keep the commandments of God and the faith of Jesus. Revelation 14:9-12
JLB
As I suspected, the phrase "if anyone" doesn't clearly or specifically identify believers. There is no reason to assume that believers will be so deceived. It's far more reasonable to conclude that those believers who have been following the beast before he forces all to take his mark will have died the sin unto death among all who will die during the judgments.
And, your logic concludes that eternal LIFE will die the second death. God's life CANNOT die. And He gives that LIFE to all who believe in His Son. And He has clearly warned of the consequences of disobedience of His children. And, nothing about His children being thrown into the fire.
I guess it is logical to conclude that what the Philistines did (fires of Molech) to their children (for 1 reason) God the Father does, for another reason. Wowsers.
Scripture is quite clear; God DETESTED child sacrifice. But your view has God doing exactly (and for eternity) what the heathens did for their own children in the fires of Molech.
And please don't compare what Jesus Christ, the Son of God, did on the cross as sacrifice for the sins of humanity. He wasn't regenerated, as human beings are. He eternally existed, and became flesh and dwelt among us. Cannot compare His sacrifice with the fires of Molech, or the second death.
Those who end up in the LoF never had eternal life. If they had, they could NOT end up there. Unless you can prove beyond any doubt that ETERNAL life can die.