How do proponents of Perseverance of the Saints have confidence in their salvation?

It all depends on how you view Scripture.

I myself am a proponent of the doctrine of the Perseverance of the Saints and I am also at the same time a proponent of the assurance/confidence of salvation.

Your premises:

"Thus, that person p cannot have confidence in their salvation."

"Thus, PoS is false."

Are based on a Hermeneutic or principle of interpretation that can be debunked if Scripture is allowed to speak. Yet, of course since there is considerable debate on the doctrine of the Perseverance of the Saints, my hermeneutic could also appear faulty.

However, I find the Doctrine of the Perseverance of the Saints to be convincing and true according to Scripture.

We read of God's keeping power:

"Oh, love the Lord, all you His saints! For the Lord preserves the faithful, And fully repays the proud person." Psalm 31:23

We may assume the following: "The Lord only preserves the faithful, but if a believer stops being faithful to God, falls away and is eternally lost, then God's keeping power is only sufficient so far as we hold on to Him."

There is truth in that statement, because Jesus said:

But he who endures to the end shall be saved. Matthew 24:13

& Paul said:

"And you, who once were alienated and enemies in your mind by wicked works, yet now He has reconciled 22 in the body of His flesh through death, to present you holy, and blameless, and above reproach in His sight— 23 if indeed you continue in the faith, grounded and steadfast, and are not moved away from the hope of the gospel which you heard, which was preached to every creature under heaven, of which I, Paul, became a minister." Colossians 1:21-23

We can't ignore the fact that believers are to endure in their faith until the end.

Yet, there are other elements of Salvation to consider.

1. Christian's must keep their faith in Christ until death in order to keep eternal life. (Scripture presented above^)
2. God is involved in every believers life, keeping them by His power through faith unto salvation:

who are kept by the power of God through faith for salvation ready to be revealed in the last time. 1 Peter 1:5

So we have to consider both the Divine element of salvation/perseverance and the Human element of it.

If salvation were initiated by us first at conversion, maintained solely by us through life, and God were to be pleased with us alone having the choice to maintain sanctification and perseverance, then we of course would conclude that "The Perseverance of the Saints" would be false. (Given the idea that we are talking about "ALL saints" here, past, present, future).

Paul Washer remarked:

"First, we must comprehend something of the nature of salvation. The superficiality of much of contemporary preaching has led us to believe that the work of salvation is primarily a decision of the human will. God reveals the gospel to man and then awaits man's response. Man receives salvation by deciding for Christ and continues in that salvation by the same act of will. Thus, the man who gains salvation through a proper response to God may just as easily lose his salvation if he renounces his initial decision by a contrary act of the will."

He continues:

The problem with this view is that it deals only with the will of man and fails to consider God's work upon his nature. The Scripture teaches that the man who believes unto salvation has been born of God. Furthermore, it affirms that this new birth is "not of blood, nor of the will of the flesh, nor of the will of man, but of God." (John 1:13).

From: Gospel Assurance & Warnings by Paul Washer, Pg 89.

To assume that God hasn't worked enough within the soul of a believer to keep him or her is to neglect certain portions of Scripture on this matter.

When Jesus gives the threat to the church in Revelation 3:5 for example, we read:

"He who overcomes shall be clothed in white garments, and I will not blot out his name from the Book of Life; but I will confess his name before My Father and before His angels." Revelation 3:5

We see this as a real and sober warning, yet with respect to the nature of the new birth there is hope and efficacious redemption already being worked out, for the Scripture says:

"For whatever is born of God overcomes the world. And this is the victory that has overcome the world—our faith. 5 Who is he who overcomes the world, but he who believes that Jesus is the Son of God?" 1 John 5:4-5

The Greek term for whatever is πᾶς (pas) meaning:all, every (thing, one), whole; always.

Source: https://www.billmounce.com/greek-dictionary/pas

So the point is that despite the real warnings, we have the assurance that all who are born of God will overcome by faith. For it is God who keeps them until the end: (Jude 1:1, 1 Peter 1:5)

For the controversial passage of Hebrews 6:4-8 being a prime argument against perseverance, we have to consider that Verse 9 concludes that to those whom the author is writing, that they can have confidence that they won't fall away "though we speak in this manner" V.9, and that their redemption is secure despite the clear warning presented.

Hebrews 6:4-8 is a warning for all Christians and the words used are obvious. Those who fall away from Christ cannot be restored to genuine repentance since they would in effect re-crucify Christ. Those who are genuinely converted, if they did fall away, will never have any chance of restoration to salvation.

"But, beloved, we are confident of better things concerning you, yes, things that accompany salvation, though we speak in this manner." Hebrews 6:9

As far as assurance of your personal salvation goes, that is contingent on what denomination or church doctrine you espouse. The Scriptures call us to make certain of our calling and election:

"Therefore, brethren, be even more diligent to make your call and election sure, for if you do these things you will never stumble; 11 for so an entrance will be supplied to you abundantly into the everlasting kingdom of our Lord and Savior Jesus Christ." 2 Peter 1:10-11

How can we be certain of our calling and election? By asking the Holy Spirit to bear witness with our spirit, that we may know we are children of God:

"For you did not receive the spirit of bondage again to fear, but you received the Spirit of adoption by whom we cry out, “Abba, Father.” 16 The Spirit Himself bears witness with our spirit that we are children of God" Romans 8:15-16

We can have assurance of salvation by prayer, so far as we are not grieving the Holy Spirit(Ephesians 4:30).

Lastly, those who professed to know Christ but who later repudiated Him are generally regarded as "false converts" primarily in reformed circles, most Christians would assume they were once true believers.

John writes:

"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

We have more examples from Scripture as well:

We read of those of whom it was said:

"This charge I commit to you, son Timothy, according to the prophecies previously made concerning you, that by them you may wage the good warfare, 19 having faith and a good conscience, which some having rejected, concerning the faith have suffered shipwreck, 20 of whom are Hymenaeus and Alexander, whom I delivered to Satan that they may learn not to blaspheme." 1 Timothy 1:18-20

Were these people, named Hymenaeus and Alexander, really saved?

Most would assert "Yes", yet when we read deeper into the Pastoral epistles we come across something that might indicate a proper exegetical foundation that shows they never were true "believers".

"And their message will spread like cancer. Hymenaeus and Philetus are of this sort, 18 who have strayed concerning the truth, saying that the resurrection is already past; and they overthrow the faith of some. 19 Nevertheless the solid foundation of God stands, having this seal: “The Lord knows those who are His,” and, “Let everyone who names the name of Christ depart from iniquity.”

20 But in a great house there are not only vessels of gold and silver, but also of wood and clay, some for honor and some for dishonor." 2 Timothy 2:17-20

The man in 2 Timothy named Hymenaeus might in fact be the one and same as in 1 Timothy 1:20. If so, then Paul makes it clear that since they "strayed concerning the truth" then we can categorize them as unsaved since Paul makes it clear that "The Lord knows those who are His" V.19.

Therefore, assuming such contextual support from both Epistles to Timothy, we can assert that those who belong to God never perish, see (John 10:27-30)

The words of Jesus should comfort us here:

"All that the Father gives Me will come to Me, and the one who comes to Me I will by no means cast out. 38 For I have come down from heaven, not to do My own will, but the will of Him who sent Me. 39 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. 40 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.” John 6:37-40

NOTE: 2 Peter 2:20-22 can be interpreted in more than one way, especially in a way that shows that believers can stumble horribly and yet recover.

CONCLUSION: Your premises cannot necessarily prove that perseverance of the saints is invalid. The proof of fruit is that only Christians can bear good fruit since they are “good tree’s”:

Even so, every good tree bears good fruit, but a bad tree bears bad fruit. A good tree cannot bear bad fruit, nor can a bad tree bear good fruit. Matthew 7:17-18

Good tree’s(true Christians) cannot bear bad fruit according to Jesus, this is in response to what you said about fruit bearing.

Further Scripture for reflection:

(Jude 24-25, John 17:1-26, Philippians 1:6, Jeremiah 32:40, 1 Peter 1:3-9, 1 Corinthians 1:4-9, 1 Thessalonians 5:23, Psalm 145:20)