A way to well-order real line How is well-ordering in real line possible?
I know that the axiom of choice provides possible well-ordering, but intuitively, this does not seem to make sense.
How can you compare 1.111111.... and 1.111111... when we can never know the full digits of each number?

I'm not sure you understand what well-ordering is. It has very little to do with comparing, in the sense of the usual ordering a set may have. For example, the usual ordering of the rationals is not a well-ordering, but one can explicitly construct a well-ordering of the rationals which bears essentially no relation to the usual ordering. 
