0.999999.... = lim(x->inf) 9/(10) + 9/(100) + 9/(1000).... 9/(10^x), 1 - 0.9999999.... = lim(x->inf) 1/(10^x) = infinitesimal number.
>In mathematics, an infinitesimal or infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero.
>but that is not zero.
the difference between 1 and 0.99999... is nonzero, therefore they are not equal. QED