that's what i meant with: "normaly you would use the things from above to proove it"
the inverse element means: a^-1*a=a*a^-1=1, in the case of 3 its just: 3*3^-1=1 and 3^-1=1/3 so
3/3=1/3*3=1 so it is proven, just by using definitions. I mean everyone wouldn't argue about this. the only thing which needs to be proven here is that the reel numbers (using +,-,*,/) is a body. this isn't that easy but can be found in many mathbooks about "Mathematical analysis 1". by multiplying the whole thing with 10 and u are at the problem of this topic.
the problem is just our imagiation of infinity (in this case infinity numbers of 3 or 9): so 0.(3) and 0.(9), the thing is, this number has no end this is the biggest issue with the decimal writing. I mean it's not a problem of left to right or right to left, as some play say here it's just an understanding of infinity
and that 1/3=.(3) seems to be known by everyone, so you can always go to the equation 3/3=1/3*3=1 and replace 1/3 with .(3): 3*.(3)=.(3)*3=1