So how can I convince him that there is such thing as Purgatory?
You probably can’t. I assume that this person is a noncatholic Christian, and if so is using the 66 book canon of the bible. This canon does not contain one of the books and passages that is used to justify prayer to and for the dead.
About all you can do is provide the evidence and let the Holy Spirit do the rest.
I would suggest you begin by reviewing this part of the
Catechism III. THE FINAL PURIFICATION, OR PURGATORY
1030 All who die in God’s grace and friendship, but still imperfectly purified, are indeed assured of their eternal salvation; but after death they undergo purification, so as to achieve the holiness necessary to enter the joy of heaven.
1031 The Church gives the name Purgatory to this final purification of the elect, which is entirely different from the punishment of the damned.606 The Church formulated her doctrine of faith on Purgatory especially at the Councils of Florence and Trent. The tradition of the Church, by reference to certain texts of Scripture, speaks of a cleansing fire:
As for certain lesser faults, we must believe that, before the Final Judgment, there is a purifying fire. He who is truth says that whoever utters blasphemy against the Holy Spirit will be pardoned neither in this age nor in the age to come. From this sentence we understand that certain offenses can be forgiven in this age, but certain others in the age to come.
1032 This teaching is also based on the practice of prayer for the dead, already mentioned in Sacred Scripture: “Therefore [Judas Maccabeus] made atonement for the dead, that they might be delivered from their sin.” From the beginning the Church has honored the memory of the dead and offered prayers in suffrage for them, above all the Eucharistic sacrifice, so that, thus purified, they may attain the beatific vision of God. The Church also commends almsgiving, indulgences, and works of penance undertaken on behalf of the dead:
Let us help and commemorate them. If Job’s sons were purified by their father’s sacrifice, why would we doubt that our offerings for the dead bring them some consolation? Let us not hesitate to help those who have died and to offer our prayers for them.
Then to support this there are these passages from Scripture.
This first is present in both the Catholic and Protestant Bibles.
I Corinthians 3:10-15 According to the grace of God which is given unto me, as a wise masterbuilder, I have laid the foundation, and another buildeth thereon. But let every man take heed how he buildeth therupon. For other foundation can no man lay than that is laid, which is Jesus Christ. Now if any man build upon this foundation gold, silver, precious stones, wood, hay, stubble;
Every man’s work shall be made manifest: for the day shall declare it, because it shall be revealed by fire; and the fire shall try every man’s work of what sort it is. If any man’s work abid which he hath built thereupon, he shall receive a reward.
If any man’s work shall be burned, he shall suffer loss: but he himself shall be saved; yet so as by fire.
This next is in the Catholic Bible but was removed from the Protestant Bibles when they removed 7 books during the Reformation in the 16th century.
The most definitive evidence of the reality of Purgatory is found in 2 Macabees 12:44-46. In this passage Judas Macabee and his men had found, in the bodies of their fallen comrades, “tokens of the idols of Jamnia, which the Law forbids the Jews to wear.” As a result, “they turned to prayer, beseeching that the sin which has been committed might be wholly blotted out…for if he were not expecting that those who had fallen would rise again, it would have been superfluous and foolish to pray for the dead…Therefore he made atonement for the dead, that they might be delivered from their sin.” This passage shows the practice and attitude of the Jews before the time of Jesus. They PRAYED for their dead and still do today.
(Source)
Finally I would simply tell him that “Purgatory” is only a dscriptive term for the place or state of being during which our “works” are tried “as by fire”, as indicated in 1 Corinthians
I hope this helps.
Peace
James