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So, this makes sense and all, but this is my question:

How do you know that 1.230000... = 1.230000....?

I guess that my real question is that if

a1.a2a3a4...aN = d1.d2d3d4...bN (finite number of digits)

then all the corresponding digits are the same.

I thought that this was true, but then again, I thought .999... < 1.000.

So. Is it true? Is it provable in the same way .999... = 1? If so, what is it?