Absolutely. Once they are regenerate of the heart (born again of the Spirit), they are kept in
God's power to the end:
In him you also, when you heard the Word of truth, the gospel of your salvation, and believed in Him, were sealed with the promised Holy Spirit, Who is the guarantee of our inheritance until we acquire possession of it, to the praise of His glory."
[Ephesians 1:13-14]
"Paul and Timothy, servants of Christ Jesus, (t)o all the saints in Christ Jesus... He who began a good work in you will bring it to completion at the day of Jesus Christ."
[Philippians 1:1-6]
"Now to him who is able to keep you from stumbling and to present you blameless before the presence of his glory with great joy, to the only God, our Savior, through Jesus Christ our Lord, be glory, majesty, dominion, and authority, before all time and now and forever."
[Jude 24-25]
The ones who do fall from grace were never members of God's Elect:
"Children, it is the last hour, and as you have heard that antichrist is coming, so now many antichrists have come. Therefore we know that it is the last hour. 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 it might become plain that they all are not of us."
[1 John 2:18-19]
We all still sin, Christian or not, and thus we are still sinners. :) We Christians still have the sinful nature ~ what Paul calls the "old man" ~ with us, but we now have a new nature also, that of the Spirit ~ Paul calls this the "new man." Paul even acknowledges and laments this dichotomy in himself and even calls himself the chief of sinners:
"For I know that nothing good dwells in me, that is, in my flesh. For I have the desire to do what is right, but not the ability to carry it out. For I do not do the good I want, but the evil I do not want is what I keep on doing."
[Romans 7:18-19]
"Christ Jesus came into the world to save sinners, of whom I am the foremost."
[1 Timothy 1:15]
This is why the writer of Hebrews exhorts us to
"...lay aside every weight, and sin which clings so closely, and let us run with endurance the race that is set before us, looking to Jesus, the founder and perfecter of our faith, Who for the joy that was set before Him endured the cross, despising the shame, and is seated at the right hand of the throne of God" [12:1-2]. As Christians, we are
redeemed sinners in this life, but yet still sinners ~ no better than anyone else ~ but loved by God in a particular way and predestined to be conformed to Christ:
"Blessed be the God and Father of our Lord Jesus Christ, who has blessed us in Christ with every spiritual blessing in the heavenly places, even as He chose us in Him before the foundation of the world, that we should be holy and blameless before Him. In love He predestined us for adoption to Himself as sons through Jesus Christ, according to the purpose of His will, to the praise of His glorious grace, with which He has blessed us in the Beloved. In Him we have redemption through His blood, the forgiveness of our trespasses, according to the riches of his grace, which He lavished upon us, in all wisdom and insight making known to us the mystery of His will, according to His purpose, which He set forth in Christ as a plan for the fullness of time, to unite all things in Him, things in heaven and things on earth."
[Ephesians 1:3-10]
By the way, this is what it means for us to be saints; not that we are now, like, "extra-special-good" or anything, but that we are sinners who have been redeemed by the Lord.
Grace and peace to you both.