no you are wrong:
since 3.3x3 = 9.9
10/3 cannot equal 3.3. and in fact you can never write 10/3 as decimal since it has infinity 3's after the dot.
and just to make u understand what you meant (even though you wrote something else):
if you take only real numbers (or complex numbers) then 9.999....=10, which can be proven by the equations you "wanted" to write. this is because the number infinity, and thus 1/infinity is not part of the real/complex numbers (even though there is no largest number, infinity does not belong to this group). even though, you can always find a real number, which is between two real numbers, you will not find anything between 9.999... and 10 (if you do tell me, than I will instantly change my opinion). the reason is easy, they are one and the same. it is easy to find that their difference is infinitily small (1/infinity), which, as we pointed out doesn't exist in the real space. so you can substract them and will obtain 0: 10-9.999...=0.
however, if you take the hyperreal numbers, things look different, this space includes infinity and 1/infinity and now these numbers are not the same anymore.
that's actually what math is: everything is proven und (mathematically proven means: it is true and it is not just a hint! there is no way that it is not true). however most proves are only done with given circumstances and thus the "rules" are only applicable for these circrumstances. thus a huge part is, to check, whether these cicrumstances are given or not. However, in your world, everything you learned is proven, and especially 9.999....=10 (which is mostly true, since hyperreal numbers are are not very applicable to reallity).