Thanks for trying to explain it to me, but unfortunately (for me) I'm still confused. It seems to me that the set of real numbers between 1 and 2 is infinite only because the number of digits to the right of the decimal point can be increased arbitrarily so that the set is countably infinite in terms of the number of digits required to represent it. I guess I am assuming (perhaps wrongly) that if an infinite set is composed of a number of countably infinite sets it itself is countably infinite. (See, I told you this wraps me around the axel.) And, of course, my thinking could be totally wrong. If you feel up to it, please teach me more.