proving something does not exist, adam slove...

It depends on what you want to prove doesn't exist and whether it can be proven not to exist.

1. You can logically prove some things don't exist, if they are necessarily false (or are based on a necessary truth). For example, logic tell us, there are no married bachelors, there are no male vixen, and there are no instances where 2+2 does not equal four.
   Localized contingent truths. You can also prove something doesn't exist by offering inspection. For example, you can prove that there are not six fingers on your hand simply by showing your hand. The person you are trying to convince can use their 5 senses to determine that a sixth finger does not exist on your hand.
   Other contingent truths. Here things can be easy or they could get a lot more difficult. Can you prove the world is not flat? There are still some people who don't believe it and won't believe you! Can you prove that Donald Trump's hair isn't actually a toupee? He says it's not. Is that good enough proof? That's going to depend on the judgment (and sanity) of the individuals you are trying to convince, isn't it? What do they require or accept as proof? So, maybe you can prove it and maybe you can't. And just maybe, you don't really need to after all.