While there are verses in the Bible that emphasize a perseverance on our part (such as Hebrews 10:36), there are also some that emphasize a perseverance on God's part. And the perseverance that God expects from us will not be accomplished unless it is He who is persevering through us. For example, Paul told believers that "it is God who works in you both to will and to do for His good pleasure" (Philippians 2:13).
Personally, I believe that God has given us a free will in order to accept His free gift of eternal life. A gift, by it's very nature, must be received - otherwise it is not truly a gift. This is why John 1:12 says "as many as received Him, to them He gave the right to become children of God". So you receive Him first, and then you become a child of God (not the other way around), which is one of the main reasons that I believe in free will.
However, although I believe in free will, I also believe that our wills are regenerated once we have been born again, and it is then God who works in us to will and to do for His good pleasure (as Philippians 2:13. declares). This is why Philippians 1:6 declares that He will complete the work that He has started in us - because it is in fact the Holy Spirit who is persevering through us!
Before telling the Philippians that it is God who works in them to will and to do for His good pleasure, Paul told them in the verse just before that to "work out your own salvation with fear and trembling" (Philippians 2:12). Notice that Paul was not telling them to work for their salvation (which is the idea that some people seem to get when they read that verse), but He was telling them to work out the salvation that God had already given them. This is also made clear by the statement "created in Christ Jesus unto good works" (Ephesians 2:10).
In other words, it is our being created in Christ Jesus that enables us to do the good works that He requires of us - and by doing them, we are working out the salvation that He has given us. This would explain what Paul meant when he said that he was "striving according to [God's] working which works in me mightily" (Colossians 1:29). To strive without His working in us will only leave us burned out, and it may not be for the right reasons anyway. But God's working in us gives the right motives as He continually searches our heart (See Psalm 139:23-24), and He also gives us the strength to do it. There are times when we can get so carried away in the work of the Lord that we forget the Lord of the work, and I say that not to look down on anyone, but because I have been there myself (which is why I know it to be true).
But the Lord's working through us is done more and more as we abide in His word: "If you abide in My word, you are my disciples indeed. And you shall know the truth, and the truth shall set you free" (John 8:31-32). It is because of this (and also because of the fact that 1 Peter 1:23 tells us that we are born again by the word of God) that I can not emphasize the importance of the word of God enough. It is what keeps us from sin (See Psalm 119:11), what gives us strength to will and to do God's pleasure, was described by a Psalmist as "a lamp unto my feet, and a light unto my path" (Psalm 119:105), and is understandable to the young (See Psalm 119:9).
And the reading of the word causes us to want to read it more. Like many other things, you need to develop a taste for it. And once you delight in His word (The Lord and His word go together), He then gives you the desires of your heart (See Psalm 37:4). Until then, we will end up frustrated seeking things that only satisfy temporary, and then leave us even more empty than we were in the first place. Once again, I have been there myself. And the times that I have been the most depressed and discouraged are times when I have not been abiding in His word.
Since it is through His word that the truth sets us free (See John 8:31-32), it is through His word that God's Holy Spirit perseveres through us - and it is in that case that "tribulation produces perseverance; and perseverance, character; and character hope" (Romans 5:3-4) - but we become burned out if we are striving in the flesh to persevere. What is needed to persevere as God would have us do so is to recognize the believers position in Christ.
As one example of this, Hebrews 10:14 says, "by one offering [Christ] has perfected forever those who are being sanctified". Obviously, someone who has been perfected forever can not be unperfected, which brings me to the words of Philippians 1:6 again: "He who has begun a good work in you will complete it until the day of Jesus Christ". With some of us, there may be missed opportunities when we could have grown more in our walk with the Lord, but the above verse guarantees that God Himself will continue to persevere in those who have "been born again, not of corruptible seed but incorruptible, through the word of God which lives and abides forever" (1 Peter 1:23). Anything that has been born of corruptible seed can parish, but those who are truly born of incorruptible seed "shall never perish" (John 10:28)! Never means never, and this is why 1 John 3:9 states that God's seed remains in those who have been born of God.
Although we will not reach sinless perfection while we are on this earth, those whom God is persevering through no longer practice sin, because they see it as God sees it. So It is then the Holy Spirit who sees it for what it is, through us, and therefore He who perseveres through us. I believe that this is what is meant by the statement, "sealed [by the Holy Spirit] unto the day of redemption" (Ephesians 4:30). A true believer may struggle with sin, (especially if they are a babe in Christ - See 1 Corinthians 3:1), but nevertheless, "sin shall not have dominion over you [if] you are no longer under law but under grace" (Romans 6:14). And once again, the word of God is where to turn if you are struggling with sin (See Psalm 119:11). When a true believer understands that they are in a position where he or she "shall never perish" (John 10:28), this gives the believer far more gratitude and a much higher motive to obey the Lord than it would to strive in the flesh to obey His commandments, which he or she then realizes "are not burdensome" (1 John 5:3). If someone who did seem to be a believer comes to a point where it is clear that he or she has rejected Christ, the following verse explains why....
"They went out from us, but they were not of us; for if they had been of us, they would have continued with us; but they went out that they might be made manifest, that none of them were of us" (1 John 2:19).