IF quotes are from Amy Carmichael, who was a Protestant Christian missionary that served in India for 55 years without furlough and wrote many books about the missionary work there.
Jesus was promised by God to "save His people from their sins." We see this with the first mention of salvation in the Bible:
Genesis 3:15 NKJV - (God speaking to the serpent) - "And I will put enmity between you and the woman, and between your seed and her seed. He shall bruise your head, and you will bruise His heel." This "Seed" would be fulfilled in Jesus.
Matthew 1:21 NKJV - (The Angel speaking to Joseph) - "And she will bring for a Son, and you shall call His name Jesus, for He will saved His people from their sins."
Before Jesus was born there were several other children who were promised to fulfill God's promise in Genesis 3:15. (This is not an exhaustive study).
Genesis 4:25 NKJV - "And Adam knew his wife again and she bore a son and named him Seth. 'For God has appointed another seed for me instead of Abel..." Seth was the ancestor of Noah, or Abraham, and ultimately, of Jesus. Seth was the next child of Eve to establish the genealogical line to Jesus.
Genesis 17:19 NKJV - "Then God said:...Sarah your wife shall bear a son, and you shall call his name Isaac. I will establish My covenant with Him..." Isaac was a direct descendant of Abraham and would establish the Messianic line to Christ.
Genesis 25:21 NKJV - "Now Isaac pleaded with the Lord for his wife, because she was barren; and the Lord granted his plead."
The child promised was Jacob who would continue the Messianic line.
II Samuel 7:12 NKJV- (God speaking to King David) - "When your days are fulfilled and you rest with your fathers, I will set up your seed after you, who will come from your body, and I will establish His kingdom forever." Solomon was the first child to bring this kingly line to Jesus.