Whereas the Catholic Church, instructed by the Holy Ghost, has from the Sacred Scriptures and the ancient tradition of the Fathers taught in Councils and very recently in this Ecumenical synod (Sess. VI, cap. XXX; Sess. XXII cap.ii, iii) that there is a purgatory, and that the souls therein are helped by the suffrages of the faithful, but principally by the acceptable Sacrifice of the Altar; the Holy Synod enjoins on the Bishops that they diligently endeavor to have the sound doctrine of the Fathers in Councils regarding purgatory everywhere taught and preached, held and believed by the faithful (Denzinger, Enchiridon, 983).
The teaching of the Magisterium is straightforward. However, it is not the only proof for the Catholic. The Magisterium is what theologians call an “Organ of Sacred Tradition.” In proof of the existence of purgatory you could also quote what theologians call the “Witnesses of Tradition,” which are equally infallible and which are represented by the unanimous consensus of the Fathers, of the theologians, of the faithful, throughout the centuries. These can be easily known through the writings of the Fathers and Doctors of the Church and of the Saints, as well as through catechisms, liturgical texts, sacred places, etc. All of Catholic culture cries out that there is indeed a Purgatory.But, practically speaking, this will hardly do for those who call themselves “Catholic” but do not believe in the authority of the Magisterium. So, if they don’t believe in Sacred Tradition (that is, neither in the Organs nor in the Witnesses of Tradition), you should at least be able to quote the other source of Divine Revelation, namely, Sacred Scripture.The most obvious text of Scripture for this purpose is 1 Maccabees 12:42-46:
And so betaking themselves to prayers, they besought him, that the sin which had been committed might be forgotten. But the most valiant Judas exhorted the people to keep themselves from sin, forasmuch as they saw before their eyes what had happened, because of the sins of those that were slain. And making a gathering, he sent twelve thousand drachms of silver to Jerusalem for sacrifice to be offered for the sins of the dead, thinking well and religiously concerning the resurrection, (For if he had not hoped that they that were slain should rise again, it would have seemed superfluous and vain to pray for the dead,) And because he considered that they who had fallen asleep with godliness, had great grace laid up for them. It is therefore a holy and wholesome thought to pray for the dead, that they may be loosed from sins.
Now, if you are speaking to a Protestant, he most likely does not believe that the books of the Maccabees are authentic books of the Sacred Scripture (these, and other books of the Old Testament are missing from their Protestant bibles). So you just quote Our Lord Himself (Mt. 12:31):
Therefore I say to you: Every sin and blasphemy shall be forgiven men, but the blasphemy of the Spirit shall not be forgiven. And whosoever shall speak a word against the Son of man, it shall be forgiven him: but he that shall speak against the Holy Ghost, it shall not be forgiven him, neither in this world, nor in the world to come."