Let's separate the idea of 'proving' from the idea of 'sourcing' - sources contribute to proof.
Every 'fact' *should* ideally have a source... a document, webpage, book or magazine, the person who witnessed the event, or the one who told you about it (you might even say something like 'family lore' as remembered by you if you can't remember who specifically told it to you). When there is conflicting information (you or others find new and different information) it's important to know where each piece of information came from so the credibility and accuracy of each source can be evaluated.
'Proof' is a loftier goal, for example have a look at the BCG's genealogical proof standard as reproduced on family search - it goes well beyong simply having a source.
If you are confident a fact is true you *can* add it without a source, but it is definitely not preferred.