The Holy Spirit testified through Jeremiah ( Jeremiah 31:33-34; cf. Hebrews 8:8-12), and continues to testify, the writer said, that final forgiveness meant the end of sacrifices for sin. God promised this forgiveness in the New Covenant. Consequently no more sacrifices for sin are necessary. [source][source][source]
"In Ch8 the oracle of Jeremiah 31:31-34 was quoted in order to prove the obsolescence of the old economy; now it is quoted again in order to establish the permanence of the era of "perfection" inaugurated under the new covenant. "God has spoken in His Son"; and He has no word to speak beyond Him." [1][source]
The statement that God will remember sins no more ( Hebrews 10:17) means that He will no longer call them back to memory with a view to condemning the sinner (cf. Romans 8:1). Since God is omniscient He remembers everything, but He does not hold the forgiven sinner"s sins against him or her. This verse has been a great help to many sinners who have found it hard to believe that God really has forgiven them (cf. 1 John 2:2). [source][source][source]
The long section on the high priestly ministry of Jesus ends here ( Hebrews 7:1 to Hebrews 10:18). Priestly ministry was such an important part of old Israelite worship that the writer gave it lengthy attention here. The writer showed that Jesus is a superior priest compared with the Levitical priests, and that His priesthood supersedes the Levitical priesthood. He also pointed out that Jesus serves under the New Covenant that is superior to the Old Covenant. Furthermore His sacrifice is superior to the animal sacrifices of the Old Covenant. Finally, Jesus" priesthood brings the believer into full acceptance with God, something the former priesthood could not do. Therefore the readers would be foolish to abandon Christianity to return to Judaism. Contemporary believers are also foolish to turn away from Christ and the gospel. [source][source][source]