Is "proving" different from "persuading"? Does proving lead to knowledge while persuading leads to belief? How do we "prove" that something is true? Are there some notions that we believe strongly, even though we can't prove them?

Persuading is convincing a group or single person of a belief while proving gives facts and reasons to show belief is correct. something fact based such as in science you can conduct experiments to prove something is true. there are some things many people believe in such as certain religions but you can't necessarily prove them to be true

Proving is different from persuading; one focuses on fact and why/how it is, while the other is led by bias and emotion.

Proving doesn't necessarily lead to knowledge, most of the time "proving" something is how facts are utilized when coming to a conclusion.

Persuading can lead to belief, though the act of persuasion is mostly seen in commercials, or any act of selling-

We prove things, by utilizing known facts related to the problem/questions

There are strong notions that we believe strongly in and cannot prove, it can be called Bias. An example of this can be but is not limited to: "Guns should be taken away from gun owners because guns 'kill'."