the lack of reasons to live is enough reason to die.
and any sort of reasons is really just attribution. attributions are not necessarily "real", just plausible enough to the person who made such an attribution.
so that goes to prove that reasons aren't meant to be inherently "true" or anything. just reasonable.
and for the vast majority of time, if not all the time, no one wishes to do anything they find unreasonable.
so there you have it.
you don't need to prove to yourself it's reasonable, you know it is or it isn't. although I'd happily quote Built To Spill to answer this question:
"proving's illogical, it's sometimes necessary; proving's illogical, but it's a cinch."