Mathematical proofs normally have the problem of either being too trivial or too hard to understand to talk about them in day to day conversations. One exception is the 0.999...=1 proof, most people instinctively do not believe it to be true, yet it is very easy to prove:
1/3 = 0.3333... | x3
3/3 = 0.999...
since 3/3=1 our proof is done at this point. The only question is if it convinced you. There are different ways to approach this problem. A long but elegant one is looking for numbers between 0.999... and 1 to see that there are none and since this cannot be the case for 2 different numbers in the set of rational numbers 0.999... and 1 have to be the same number.
Do you believe me or did you already know and want me to do some more hardcore math proofs on my blog?