Proof of the Church’s infallibility
That the Church is infallible in her definitions on faith and morals is itself a Catholic dogma, which, although it was formulated ecumenically for the first time in the Vatican Council, had been explicitly taught long before and had been assumed from the very beginning without question down to the time of the Protestant Reformation.
http://www.newadvent.org/cathen/07790a.htm
Hell did you even read the link you posted? LCS is correct.
‘That the Church is infallible in her definitions on faith and morals is itself a Catholic dogma’
That is true, in certain limited circumstances, but it is not what you said originally.
Outside of those particular conditions, there is no claim that any one person is infallible. And certainly, there is no claim at all that any clergy or laity are without sin - there have been many sinful priests, bishops, and popes (to the shame of Christ and His Church) just as there have been many holy ones.
This has been going on since the beginning of the Church: even Jesus chose a Judas Iscariot who betrayed Him, a Simon Peter who denied Him, a Thomas who doubted that He was resurrected - and the other nine weren't without their own shortcomings either.
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But Jesus preached a message of repentance and mercy: Jesus forgave Peter's denial and restored him to the position of feeding His flock; Thomas went on to preach Christ Crucified and Resurrected in India and was martyred for the Faith, and the others went on to preach the Gospel as well. The opportunity is there for people to become great Saints - we simply don't always take it.