There are also more advanced methods of proving 0.999…=1, listed here on Wikipedia. (http://en.wikipedia.org/wiki/0.999…) Some of the techniques include Infinite series and sequences, Dedekind cuts, and Cauchy sequences.
There are always two points on opposite sides of the Earth with the exact same temperature. And we can prove that.
Temperature changes continuously. If a and b are on opposite sides of the equator and D(a) = T(a) – T(b) is positive, then D(b) = T(b) – T(a) is negative. That means there must be some point x on the equator where D(x) = 0. At that point the two opposite sides are the same temperature.
Mathematicians call this the Intermediate Value Theorem which means if there is a continuous function that changes from of a positive value to a negative value (or the other way around) then it must, at some point, pass through zero.