Succession as a principle was understood and carefully maintained within the early Church. The original Apostles were also thought of as “overseers” or “bishops”. Eusebius writes the following:“All that time most of the apostles and disciples, including James himself, the first Bishop of Jerusalem, known as the Lord’s brother, were still alive…” (History of the Church, 7:19, tr. G.A. Williamson, Baltimore: Penguin Books, 1965, p. 118)James is called an Apostle by St. Paul in Gal 1:19 and 1 Cor 15:7. That James was the sole, “monarchical” bishop of Jerusalem is fairly apparent from Scripture also (Acts 12:17, 15:13,19, 21:18, Galatians 1:19, 2:12).
The Twelve(+) were unique in that they had “seen the Lord”. They functioned as the senior leaders in the early church and as such would have also held the role of overseer (Bishop). As the Church expanded, the Apostles named helpers; first the deacons and later elders (presbyters/priests).
Due to the growth of the Church and the deaths of the Twelve(+) Apostles, the senior leadership of the early Church passed to the Overseers (Bishops) who were next in line as the senior leaders in the Church.
By the laying on of hands in a process known as Apostolic Succession, these men received the Apostolic Authority of the Apostles. The promises of the Lord to remain with them “forever” continues in the valid line of Bishops to this very day.
Infallibility is found in numerous passages of scripture including:
"I have much more to say to you, more than you can now bear. But when he, the Spirit of truth, comes, he will guide you into all truth. He will not speak on his own; he will speak only what he hears, and he will tell you what is yet to come. He will bring glory to me by taking from what is mine and making it known to you. All that belongs to the Father is mine. That is why I said the Spirit will take from what is mine and make it known to you."
The Original Twelve did not receive ALL that Jesus wanted to teach them - it was more than they could bear. However, their successors the Bishops of the Catholic Church have been lead “into all truth” and protected from error by the Holy Spirit.
From the Catechism of the Catholic Church:
830 The word “catholic” means “universal,” in the sense of “according to the totality” or “in keeping with the whole.” The Church is catholic in a double sense:
First, the Church is catholic because Christ is present in her. “Where there is Christ Jesus, there is the Catholic Church.” In her subsists the fullness of Christ’s body united with its head; this implies that she receives from him “the fullness of the means of salvation” which he has willed: correct and complete confession of faith, full sacramental life, and ordained ministry in apostolic succession. The Church was, in this fundamental sense, catholic on the day of Pentecost and will always be so until the day of the Parousia.