Are you trying to convince me that "the rationals are continuous?" I don't get it. You misspoke yourself, I pointed out your factual error. You should just say, "Hey thanks for the clarification," and leave it at that. I can't understand why you're trying to convince me of something you already acknowledged you're wrong about. Of course it's true that an infinite sum of rationals can converge to an irrational, but that just goes to further falsify your original point. Which you already agreed is false. I don't see this kind of behavior as conducive to rational discussion. By the way the property of a linearly ordered set that there's always an element strictly between any other two elements is density. The rationals in their usual order are a dense set.