How do we know Judas was really lost?

First, I think you are referring to Judas Iscariot (John 6:71; 13:26 ) and not Jesus' half-brother whose name was Judas (John 14:22). It is Judas Iscariot who Jesus referred to as the “son of destruction” (John 17:12). Satan entered into him (John 13:27; cf. John 6:70) and he betrayed Christ with a prearranged deceitful kiss (Mark 14:44; Luke 22:48). Judas Iscariot fulfilled Scripture (Psa. 41:9; John 13:18) and his eternal judgment was in keeping with the crime of all crimes — "It would be better for him if he had not been born" (Matt. 26:24). He was condemned to an eternal hell (see below).

Judas Iscariot, as one of the twelve (Matt. 26:14-15), was close to Christ and even did miracles in his name, but unlike the other disciples he was not truly "born again" or “clean” (John 13:10-11). However, at the end of his deceptive ministry and with his real betrayal, he was a worthy recipient of these words: "I never knew you; depart from me, you worker of lawlessness" (Matt. 7:22-23). Judas was eternally lost. This is affirmed when John wrote: "While I was with them, I kept them in your name, which you have given me. I have guarded them, and not one of them has been lost except the son of destruction, that the Scripture might be fulfilled" (John 17:12; cf. Matt. 18:7).

While Judas repented (Matt. 27:2-5), it was a false or worldly repentance (2 Cor. 7:10), not one as given by the Holy Spirit (Acts 11:18; 2 Tim. 2:24-26; cf. Rom. 2:4). He was always a seed of the Serpent (John 8:44), being both unable and unwilling to repent (Rom. 8:7-8) and to come to faith in Jesus.

Judas Iscariot was predestined to be a son of perdition (Psa. 41:9; Zech. 11:11-12; John 13:18; cf. Acts 2:23-24; 4:27-28; Rom. 9:15-16, 18, 21). This was confirmed by his life and works (Luke 22:46). All things are ordained by God, even Judas and his works. A particular circumstance doesn’t matter because no one can do anything other than what God has ordained. However, this does not mean a person is merely a robot. It means, rather, that within what he ordains, God establishes (or grants) the ability of the person’s own choosing by way of their greatest inclinations at the moment they so choose. No violence is done to one's will by the holy ordaining of God. He knows what will take place because he ordains what will take place necessarily, freely, or contingently by his own moral agency.

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