“If someone does not persevere until the end, then by definition they were not among the elect in the first place”
It was quoted accurately.
Charles Hodge (Systematic Theology, 3.16.8):
“Perseverance...is due to the purpose of God [in saving men and thereby bringing glory to his name], to the work of Christ [in cancelling men’s debt and earning their righteousness ], to the indwelling of the Holy Spirit [in sealing men in salvation and leading them in God’s ways], and to the primal source of all, the infinite, mysterious, and immutable love of God.”
After God has regenerated someone, the person’s will cannot reverse its course. The person is indwelt and led by the Holy Spirit and nothing and nobody can separate them from the love of God. God has changed that person in ways that are outside of his or her own ability to alter fundamentally, and he or she will therefore persevere in the faith. It is the seed that fell on good ground and bore much fruit (persevered). Seed fell: on hard ground and Satan takes it with no fruit; rocky ground that had some fruit but when trials came died; among thorns and bore fruit until the cares of the world choked it and died. None persevered since the seed fell in places not prepared by the sower like the good ground had been.
Jesus said that many will say “Lord, Lord” but he will say “I never knew you.”
A lot of people claim Christianity. Not all of them are.