There are three big factors at play. These factors are magnified in cults, but you can actually see them working at the small scale even in your office or family about completely trivial subjects.
The first is hierarchy - most people get their idea of true/false right/wrong through the social structure they're embedded in, and from who they consider senior and look up to. This is pretty fixed even if the senior figure gets something conspicuously wrong.
The second is consistency; beliefs usually come in sets, and they're often tied together by being supported by the same sources (see above). And it's especially tempting to have a consistent set of beliefs that happen to be wrong, rather than deal with the messy inconsistency and uncertainty of the real world. This is why people don't get partially deprogrammed from cults.
The third is loyalty and identity group membership; this is the peer-to-peer version of "hierarchy" above. Disagreement is an attack on the peer group. The extreme version of this is seen in religions who will ostracise unbelievers. If you want to stop being a Mormon or a JW you may have to walk away from everyone you've ever known.
"Sources" are fairly useless in this situation unless you can find one that's already inside the target's perception of loyalty and hierarchy. Anything from outside can be easily dismissed.