It's pretty simple to find the 'error', but a funny equation to show students
Well it can be shown that if you come up with a result that contradicts the original axioms used, then either somehow you made the axioms fail in your line of thinking, either you failed in your line of thinking. The first case hasn't been stumbled upon yet ... and I wonder if it can be proved that they can't fail *accidentally*It's good to realise though that to people experienced in mathematics, the errors in these types of "proofs" tend to stand out immediately and appear rather silly.
But yes, it's quite fun sometimes to check out those "failproofs". You can check out this link, there is a more subtile version of the same contradiction, which is quite harder to see and a lot more interesting IHMO.
Googling "Fallacious proof" also returns a lot of interesting results