Here is a great quote from J. Dwight Pentecost, writing in 1965, on why the promise of eternal life implies eternal security:

The promise of God is a basis for our security. We go into a familiar passage such as John 3:16 where it is made so clear: “…God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.” Notice the two aspects of the promise: negatively, he shall not perish; positively, he shall have everlasting life!… When God offers a man life, God offers a man only one kind of life, and that is eternal life. Eternal life is the life of God, and as God’s life could never be terminated by death, so the life of God, given to the child of God, could never be terminated. We submit to you that the promise of God to give eternal life to the one who accepts Christ as his Saviour [sic] is a sufficient basis for our security (Things Which Become Sound Doctrine, 127, emphasis added).

“When God offers a man life, God offers a man only one kind of life, and that is eternal life.” I love how Pentecost is so clear that God only promises life that cannot be lost (i.e., that is eternally secure). Anything less than that is not something promised by God.

Eternal security is not a side-benefit of salvation; it is the promise of salvation.