Forget Jesuit Futurism and get to know Protestant Historicism.
This is symbolism. A "beast" is a kingdom, not a man. The "Beast" or "Little Horn" arises after the destruction of the four other beasts, namely Babylon, Medo Persia, Greece, and Rome. This Little Horn Beast was to arise among the Ten Horns, the ten barbarian tribes which invaded and destroyed the Roman Empire, and this Little Horn was to continue until the end of time and to its appointed day of destruction (which proves it's not an individual, but a kingdom).
Now, the symbolism refers to the other beasts "who had their lives prolonged". Since we cannot interpret this literally, seeing that these empires are long gone, the only explanation is that the "philosophies" of these kingdoms is what lives on, all of which make up the characteristics of the Little Horn:
Babylon: Sun god worship
Medo Persia: irrevocable code of law demanding strict (blind) obedience
Greece: human wisdom exalted over God's Word
Rome: iron fist (and nails) persecution and death of anyone who dares oppose it
Now, the symbolism refers to the other beasts "who had their lives prolonged". Since we cannot interpret this literally, seeing that these empires are long gone, the only explanation is that the "philosophies" of these kingdoms is what lives on, all of which make up the characteristics of the Little Horn:
Babylon: Sun god worship
Medo Persia: irrevocable code of law demanding strict (blind) obedience
Greece: human wisdom exalted over God's Word
Rome: iron fist (and nails) persecution and death of anyone who dares oppose it
But the portion I marked in red isn't what the passage says. Notice the horn comes OUT OF the 4th beast and speaks, and then the 4th beast is destroyed by burning flame...this is the same destruction that we read from 2 Thessalonians 2 when the man of sin is destroyed with the brightness of the Messiah's coming.
Daniel 7:11
11 I beheld then because of the voice of the great words which the horn spake: I beheld even till the beast was slain, and his body destroyed, and given to the burning flame.
Daniel saw the pompously speaking horn UNTIL the 4th beast was destroyed by fire. After the horn (and the 4th beast it grew out of) is burned and destroyed, we're told...
Daniel 7:12
12 As concerning the rest of the beasts, they had their dominion taken away: yet their lives were prolonged for a season and time.
Now whether we argue that "dominion" was taken away by the 4th beast (Rome) or by the Messiah, we know that dominion means "world rule". So these kingdoms would still exist for a time and season. Well today, there is STILL a Babylon (Iraq), a Persia (Iran), a Greece, and a Rome (the seat of the papacy).