Does infinity exist?

So I got to thinking, why do digits extend into infinity past the decimal, is it impossible to reach an atomic number which cannot be divided. I think that asserting that there is always a .999 for  everything is making a dangerous assumption. It begs the question, does infinity exist? Of course that is relative, it is easy to have .99 dollars, or .9999999 populations of the earth, but is there a measurement of subatomic particles with a cap on the number of 9's you can have?

does infinity exist in the real world?  doesn't matter.
does infinity exist as a mathematical construct we made?  yes