Many people have trouble with the concept of Papal Infallibility simply because they don't understand what it means, or they assume it means something it doesn't. Although Fr. Baron explains it fully in the video above from the Word On Fire's Catholicism series, I'd like to touch on a few points.
First, it's necessary to know what Infallibility (Pope-wise) doesn't mean. It doesn't mean he never sins. To never sin would be impeccability, which no Pope could claim. In fact some Popes in the Catholic Church's rich past sinned an awful lot, eg., Pope Alexander VI (Borgia), who led a rather immoral and lascivious lifestyle, and is considered by many to be the most corrupt man ever to hold the office.
What Infallibility means is that the Pope is never wrong when defining doctrine regarding faith and morals (when speaking ex cathedra). You can read a much longer definition by clicking HERE, where you will also find biblical scripture to back up the concept.
But what I do is simply look more closely at Alexander VI for proof. Because a man living such an evil life in a position of enormous power (much more so in those days) would seek to validate such lasciviousness by perverting Catholic dogma to his own selfish ends so he could live out his twisted desires openly and without scorn. So why didn't he do that? Because by virtue of his position, though God allowed him (and other Popes) free will to act badly, The Holy Spirit would not allow him to pervert doctrinal teaching on faith and morals, any more than the Pope could claim 2 + 2 = 5.
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