perhaps you should ask yourself (apparently no one ever does), why:
1 = 1.00000000~ (an infinite string of zeros).
certainly, as strings of symbols, these are not the same string (they have differing lengths). it is not just "decimal representations ending in 9's eventually" that have non-unique representation. EVERY finite decimal has a non-unique representation (an infinite number of finite representations, and two infinite ones).
the reason for this peculiarity is "the decimal system", which is base 10. and 9 is just 1 (the smallest digit available) away from 10. similar problem arise if you switch to another base:
in base 3:
1 = 0.2222222222222222~
as far as "proving in reality by proving in mathematics", the situation is thus:
we have a "real problem" (perhaps an engineering problem, for example). we "abstract the situation (model it)" and re-formulate it as a mathematical problem. we then use the techniques of logical formalism math so excels at, to (hopefully) arrive at a simpler expression of our problem (the beam will carry 500 pounds per foot, for example), and then apply this to our "real world".
the question is: our are models appropriate, and accurate? and the best answer we have, is: "it depends". for some things, there appears to be a high degree of agreement between the model, and the "real situation". companies that run state lotteries, for example, use statistics and probablilty very effectively to avoid losing money (so do casinos). the utility of arithmetic in figuring such things as taxes, balancing one's checkbook, and computing how much paint to buy to paint a room is undeniable.
bolstered by such satisfying results, one wonders if it might be possible to model EVERYTHING mathematically. but here, the best minds of our species have run into some snags. the exact nature of the very large, and the very small, are not entirely known. we HOPE that such things can indeed be modelled by mathematics, and indeed many working cosmologists, and sub-atomic physicists, begin by formulating a mathematical model, and checking to see if that is a) consistent with what we know so far, and b) predictive of things we do not yet know. there have been some successes in this vein, and some believe it is the best available approach. realize, however, that you are now in the realm of philosophy, it could very well be the case, that there is something about the universe which we are not capable of modelling (perhaps not even a limit of mathematics, per se, but of our limited ability to apprehend the world).
it is not known if the universe is "bounded" (finite), or not. some theorize it is. some disagree. so while one cannot say for sure if "a completed infinity" exists in nature, one certainly does in mathematics (unless you are an ultrafinitist, which most consider an extreme and untenable position). in fact, one of the basic "axioms" of set theory (the most currently popular foundation for mathematics), is simply "an infinte set exists" (this set is often identified with the natural numbers).
mathematics is "bigger" than reality, in the sense that one can do perfectly acceptable mathematics, without any regard for some "instantiation" of it in the real world. on the other hand, we are in the world, and mathematics is in our heads, which leaves us in the peculiar position of being able to imagine we are bigger than we are. while this leads to no great crisis of faith when we play online role-playing games (imagination in fiction doesn't seem to pose the same sort of epistemological paradoxes), it does lead us to wonder about how we can think about things with such a tangled hierachy of "in".