"But beloved, we have persuaded ourselves better things of you, and such as accompany salvation, though we thus speak. For God is not unrighteous, that he should forget your work, and labor of love, which ye showed toward his Name, in that ye have ministered unto the Saints, and yet minister." - Hebrews 6:9-10 (GNV) 
In the previous passage (verses 4-8), the author of Hebrews was severely warning his original audience. As the Geneva Bible put it, He terrifieth them with severe threatenings.
The writer was afraid that writing in such a strong manner, might leave his recipients in a state of utter desperation, in which they would feel no sense of security at all any longer. Therefore the author wrote a note on the passage he wrote before. He makes clear what he wants to express. By stating: “though we thus speak”, he mitigates the severe tone of his words.
After sharing with them the fact that they are beloved, the author continues his argument by stating that “we have persuaded ourselves better things of you”. An imperfect analogy can be made; suppose a teacher gives a student feedback on a mid-term exam. The student has unfortunately failed the test. Therefore, the teacher says: “William, you have failed this test. If you continue like this, you will have to retake this course next year and you will have to pay for an additional year at this school”. The teacher continues by saying: “But I know that you can do better William! You are gifted with a very high intellectual capacity!”. Compare this imperfect analogy with the writer of the Epistle’s intent: “If you fall away from the faith (v 6), you will burn (v 8). But God knows that you have done good deeds out of love for Him (v 10). Therefore, continue doing the same acts of diligence – out of love for Jesus – so that you may remain assured in your hope for your final salvation (v 11)”. As the Geneva Bible wrote about the author of the Epistle to the Hebrews: “He mitigateth and assuageth all that sharpness, hoping better of them to whom he writeth” (footnote on verse 9). and “He praiseth them for their charity, thereby encouraging them to go forward, and to hold out to the end” (footnote on verse 10).
We can summarize this argument form:
Teacher: statement of the fact -> severe warning -> encouragement
The writer of Hebrews: severe warning -> encouragement
By writing the passage we are considering, the author relieved his original recipients of some unnecessary stress. Through their obedient heart of true faith towards God – which is accompanied by works -, they were fulfilling God’s requirement for them to have salvific security in Christ. All they had to do is continuing in this obedient walk of childlike faith. If they continue walking in faith and patience, they are marching onwards to inheriting God’s indescribable promises (v. 12-15), which are conditional (v. 4-8,… + footnotes 2 and 3 of this article + conclusion of this article), but which can be relied upon (v. 17-19). 
This moderating note to the previous passage is clearly only applicable to obedient Christians. Thus, we can conclude that this passage flows with the author’s severe warning to disobedient Christians, as found in Hebrews 6:4-8. There is absolute security for obedient Christians but this is not the case for disobedient Christians.
Click here to find out what the entire Book of Hebrews says about eternal/conditional security.
Click here to find out what the entire Book of Jude says about eternal/conditional security.
Click here to find out more about eternal/conditional security in general.
 My opinion is that the best rendering of these verses, I’ve read so far, is the Geneva Bible rendering (which I just quoted). I believe it conveys well the writer’s original intent of writing these remarks on what he previously wrote in verses 4-8.
 The author gives some examples, by which he tries to show that God’s promises can be relied upon. The heavenly bliss, with all its rewards, is waiting for them, IF they, like their ancestors, THROUGH FAITH AND PATIENCE (v. 12), ARE HOLDING FAST TO (κρατέω , v. 18) God’s oath (v. 11-12(-18)); the hope which is an anchor to their soul (v. 17-19). Jesus is the example par excellence. After His suffering on the cross, He became a high priest in the order of Melchizedek (v. 20). Therefore “he is able also perfectly to save them that come unto God by him” (Hebrews 7:25, GNV). To put it in other words; after His suffering, Jesus finally received His satisfaction; receiving many justified, forgiven and cleansed souls, given to Him by God the Father, according to Isaiah 53:11-12.
 In Greek, this verb tense implies a continued action; hence “are holding fast to”. Strong’s concordance has: “1) to have power, be powerful 1a) to be chief, be master of, to rule 2) to get possession of 2a) to become master of, to obtain 2b) to take hold of 2c) to take hold of, take, seize 2c1) to lay hands on one in order to get him into one’s power 3) to hold 3a) to hold in the hand 3b) to hold fast, i.e. not discard or let go 3b1) to keep carefully and faithfully 3c) to continue to hold, to retain 3c1) of death continuing to hold one 3c2) to hold in check, restrain”. This makes it clear that this is A CONDITIONAL STATEMENT. I would render the verse thusly: “That by two immutable things, in which it was impossible for God to lie, we might have a strong consolation, who have fled for refuge to HOLD FAST TO / KEEP CAREFULLY AND FAITHFULLY the hope set before us” (translation of this passage is based on the KJV).