Hebrews 9:11-15
When Christ came as high priest of the good things that are already here, he went through the greater and more perfect tabernacle that is not man-made, that is to say, not a part of this creation. He did not enter by means of the blood of goats and calves; but he entered the Most Holy Place once for all by his own blood, having obtained eternal redemption. The blood of goats and bulls and the ashes of a heifer sprinkled on those who are ceremonially unclean sanctify them so that they are outwardly clean. How much more, then, will the blood of Christ, who through the eternal Spirit offered himself unblemished to God, cleanse our consciences from acts that lead to death,[c] so that we may serve the living God!
For this reason Christ is the mediator of a new covenant, that those who are called may receive the promised eternal inheritance—now that he has died as a ransom to set them free from the sins committed under the first covenant.

Hebrews 8 gives us some details about the new covenant. The new covenant involves Jesus as the mediator and high priest, is "superior to the old one", and is "founded on better promises". God found fault with the old obsolete covenant because the people did not remain faithful. God said under the new covenant, He will "put My laws in their minds and write them on their hearts". And, "they will all know Me" , for "I will forgive their wickedness and will remember their sins no more". What a wonderful promise of hope to rest in for us who are under the new covenant of Jesus !

Hebrews 7 clarifies the reference in 6:20 to Melchizedek. Paul gives us some explanation of the significance of Jesus being our high priest “in the order of Melchizedek”. Melchizedek was the King of Salem , which means “king of peace” and Melchizedek means “king of righteousness”. Paul says that Melchizedek had no father or mother, no genealogy, and without beginning of days or end of life, so he remains a priest forever, just like the Son of God. Next, Paul explains that Jesus’s priesthood is superior to the levitical priesthood. They were appointed priests based on their ancestry, whereas Jesus demonstrated an “indestructible life”. Verse 6:12 says that with a change in the priesthood, there must also be a change in the law (regulations), so the former regulation is set aside “because it was weak and useless—for the law made nothing perfect”. “A better hope “ has been introduced in Jesus, by which we draw near to God. Furthermore, Paul says that Jesus has become a guarantee of a better covenant . And, since Jesus’ priesthood is forever, he is able to save completely. Jesus is holy and pure, so he was able to sacrifice for our sins once for all time when He offered himself for us, unlike the other high priests who needed to offer sacrifices day after day.

Hebrews 6 is difficult to interpret and many good bible scholars come to quite different conclusions about the details, but nearly all agree that there is no basis in this chapter for denying the biblical doctrine of eternal security. The book was written to Hebrews, who had a difficult time turning from their old Judaism performance-based faith. The lesson is to accept that Jesus paid for our sins FOR ALL TIME and move on. We don't need to fret about the possibility of God turning His back on us if our faith falters, since He remains always faithful.

In Hebrews 5, Paul continues to demonstrate Jesus and Christianity to be superior to the old priesthood and Judaism. Paul reminds us that God announced that Jesus is His Son and that Jesus is our eternal high priest. Next, Paul presents a call us to spiritual training, so that we would have the skill to recognize the difference between right and wrong. We need to listen to God and not let ourselves become spiritually dull, and God will then grow us up to be mature enough to teach others the basic things about God's Word.

Hebrews 4
For the word of God is living and active. Sharper than any double-edged sword, it penetrates even to dividing soul and spirit, joints and marrow; it judges the thoughts and attitudes of the heart. Nothing in all creation is hidden from God's sight.

Hebrews 3 is a warning to hold fast to our belief in Jesus. Paul says to "fix your thoughts on Jesus" and "hold on to our courage and the hope of which we boast". During times of testing, "if you hear His voice, do not harden your hearts", but hold firmly to your confidence in Him, so that you may "enter His rest".