Quote by Hippasos
So the chain
0<1<2<3...
1. Never ends while reaching infinity?
2. Never ends and does not ever reach infinity?

We don't really talk about reaching infinity. Infinity is a concept, not a number; you can not perform most mathematical procedures on it. (For example you cannot claim that infinity/infinity = 1)
We talk about numbers as they
approach infinity. I guess that's tantamount to your #2.