In the Book of Acts, specifically Acts 10, God convinced Peter of the inclusion of Gentiles in the offer of eternal life through a vision. In this vision, Peter saw a large sheet being lowered from heaven, containing various unclean animals, and he was instructed to kill and eat. When Peter hesitated, feeling it was against Jewish law, God told him that what He has made clean should not be considered unholy. This vision symbolized the breaking down of barriers between Jews and Gentiles, indicating that the Gospel was for all people, not just the Jewish nation.
Additionally, it was through the events that unfolded with Cornelius, a Gentile centurion, who received a message from an angel telling him to send for Peter. Peter's encounter with Cornelius and the Holy Spirit’s descent upon the Gentiles who believed reinforced the message from God. Thus, the combination of the vision and the subsequent events led Peter to understand that God offers eternal life through Jesus Christ to both Jewish people and Gentiles.