Based on various sources (the articles Perseverance of the Saints, p. 197 in Essential Truths of the Christian Faith by R.C. Sproul; the video What Is Reformed Theology? by R.C. Sproul; Grace Unknown (chapter 10) by R.C. Sproul).
Can you lose your salvation?
persevere - to persist in a state, enterprise, or undertaking in spite of counterinfluences, opposition, or discouragement
The word persevere seems to indicate that it is us who perseveres until the end, leaving the work to us. Perhaps it would be better to think of it as Preservation of the Saints.
preserve - 1) to keep safe from injury, harm, or destruction 2) to keep alive, intact, or free from decay 3) to keep or save from decomposition
God preserves His saints.
Since we have studied total depravity, unconditional election and irresistible grace, I think it is easier to understand the perseverance (or preservation) of the saints. It seems logical that if it was not us who began the work of salvation, then it could not be us who continues the work of salvation. Scripture says:
Moreover whom He predestined, these He also called; whom He called, these He also justified; and whom He justified, these He also glorified. What then shall we say to these things? If God is for us, who can be against us? He who did not spare His own Son, but delivered Him up for us all, how shall He not with Him also freely give us all things? Who shall bring a charge against God's elect? It is God who justifies. Who is he who condemns? It is Christ who died, and furthermore is also risen, who is even at the right hand of God, who also makes intercession for us. Who shall separate us from the love of Christ? Shall tribulation, or distress, or persecution, or famine, or nakedness, or peril, or sword? As it is written: "For Your sake we are killed all day long; we are accounted as sheep for the slaughter." Yet in all these things we are more than conquerors through Him who loved us. For I am persuaded that neither death nor life, nor angels nor principalities nor powers, nor things present nor things to come, nor height nor depth, nor any other created thing, shall be able to separate us from the love of God which is in Christ Jesus our Lord. (NKJ)
Paul and Timothy, bondservants of Jesus Christ, To all the saints in Christ Jesus who are in Philippi, with the bishops and deacons: Grace to you and peace from God our Father and the Lord Jesus Christ. I thank my God upon every remembrance of you, always in every prayer of mine making request for you all with joy, for your fellowship in the gospel from the first day until now, being confident of this very thing, that He who has begun a good work in you will complete it until the day of Jesus Christ; (NKJ)
But Christ came as High Priest of the good things to come, with the greater and more perfect tabernacle not made with hands, that is, not of this creation. Not with the blood of goats and calves, but with His own blood He entered the Most Holy Place once for all, having obtained eternal redemption. For if the blood of bulls and goats and the ashes of a heifer, sprinkling the unclean, sanctifies for the purifying of the flesh, how much more shall the blood of Christ, who through the eternal Spirit offered Himself without spot to God, cleanse your conscience from dead works to serve the living God? And for this reason He is the Mediator of the new covenant, by means of death, for the redemption of the transgressions under the first covenant, that those who are called may receive the promise of the eternal inheritance. (NKJ)
And Jesus said to them, "I am the bread of life. He who comes to Me shall never hunger, and he who believes in Me shall never thirst. "But I said to you that you have seen Me and yet do not believe. "All that the Father gives Me will come to Me, and the one who comes to Me I will by no means cast out. "For I have come down from heaven, not to do My own will, but the will of Him who sent Me. "This is the will of the Father who sent Me, that of all He has given Me I should lose nothing, but should raise it up at the last day. "And this is the will of Him who sent Me, that everyone who sees the Son and believes in Him may have everlasting life; and I will raise him up at the last day." (NKJ)
Now it was the Feast of Dedication in Jerusalem, and it was winter. And Jesus walked in the temple, in Solomon's porch. Then the Jews surrounded Him and said to Him, "How long do You keep us in doubt? If You are the Christ, tell us plainly." Jesus answered them, "I told you, and you do not believe. The works that I do in My Father's name, they bear witness of Me. "But you do not believe, because you are not of My sheep, as I said to you. "My sheep hear My voice, and I know them, and they follow Me. "And I give them eternal life, and they shall never perish; neither shall anyone snatch them out of My hand. "My Father, who has given them to Me, is greater than all; and no one is able to snatch them out of My Father's hand. "I and My Father are one." (NKJ)
Does all this mean that we will never fall into sin? Certainly, not. Great men of faith, such as King David and the apostle Peter fell into terrible sin. Did they repent? Were they restored? You bet! Were they ever lost or in danger of hell? No way! The Westminster Confession says: