From Anstey’s Doctrinal Definitions:
The exact phrase “eternal security” is not found in the Bible, but the truth that it conveys certainly is there. It refers to the fact that if a person is truly saved through faith in Christ, he cannot lose his soul’s salvation and acceptance in Christ before God. The following verses teach this: Luke 15:3-6; John 6:37-40; 10:28-29; 14:16; Romans 6:23 with 11:29; Romans 8:30-39; 1 Corinthians 1:7-8; 3:13-17; 5:5; Ephesians 1:13 with 4:30; Philippians 1:6; 1 Timothy 4:1; Hebrews 10:14; 13:5; 1 Peter 1:5; 1 John 2:1.

If a believer were to lose his salvation and end up in Hell, God would prove to be a liar, because His Word says that Christ’s sheep will “never perish” (John 10:28-29). Christ would have to end up in Hell with the believer, because He promised: “I will never leave thee, nor forsake thee” (Heb. 13:5). The Holy Spirit would also have to go to Hell, because Scripture says that He will dwell in and with believers “forever” (John 14:16). All this, of course, is ludicrous and totally impossible.

There are some passages of Scripture that seem to teach that a believer could lose his salvation, but a closer look at these passages shows that they are not speaking of real believers on the Lord Jesus Christ, but rather of merely professing believers who apostatize from the Christian faith. Some of these passages are: Matthew 7:21-23; 12:43-45; 13:5-6, 20-21; 24:13; 25:26-30; Mark 3:28-30; Luke 22:31-32; John 15:2-6; Romans 11:22; 1 Cor. 9:27; 15:2; Hebrews 6:4-6; 10:26-29; 12:14; 2 Peter 2:1, 20-21. The difficulty people have with these passages—that leads them to a wrong conclusion—is that they don’t know the difference between backsliding and apostasy. Both of these things refer to a person’s departure from God, but one (apostasy) is infinitely worse than the other. A real believer may backslide and get away from walking with the Lord, but he will not apostatize. Only merely professing believers can apostatize, which is to abandon the confession of faith that they once made. If they do this, they are damned, even though they’re still alive in this world! They cannot be renewed to repentance and brought to salvation through faith in Christ (Heb. 6:4-6; 10:26).

The key to understanding these passages that seem to teach that a believer could lose his salvation is to see that they are referring to apostasy, not backsliding. They have to do with merely professing believers, not real believers. It may be asked, “Why would these warnings concerning apostasy be stated in books of the Bible that were written to believers when it has no application to them?” The answer is that the divinely inspired writers of the New Testament were on many occasions addressing a mixed multitude of real and merely professing believers. Thus, their remarks included warnings for those moving among the real believers who were merely professing faith in Christ. Such remarks were intended for the consciences of these people; they were intended to awaken them to their need to get saved, and to warn them that if they turned their backs on the faith they were professing, they would be lost forever! (See Apostasy and Backsliding.)