Sec. 7, first half of 10.3, due March 12

The main thing I can't understand is the proof by contradiction used to show that (0,1) is an uncountable set.  Somehow they took a1=0.31717..., a2=0.1515..., and a3=.04000....  Because of this it said that the first three digits in the decimal expansion of b are 5, 4, 5, or b=0.545.  Anyway, more explanation of the proof would be helpful.  Once you're able to assume that theorem it should become significantly easier to prove that something is uncountable.