> So the zero width area starting at 3/8, non-inclusive. So yeah, is that just the interval (3/8, 3/8), i.e. the empty set, or what? If it were inclusive, yo

> So the zero width area starting at 3/8, non-inclusive. So yeah, is that just the interval (3/8, 3/8), i.e. the empty set, or what? If it were inclusive, you'd have [3/8, 3/8], which means the same thing as {3/8} -- it's the set that contains just the value 3/8. > Is it meaningful to talk about rounding reals to the nearest rational? If not, why not? It is not. The reason is, there is no nearest rational (if it's not already rational). Suppose x is an irrational number, and p is some rational number. Then p != x. That means |x-p| > 0. That means there's some integer n such that 1/n < |x-p|. If p > x, then x < (p - 1/n) < p. If p < x, then p < (p + 1/n) < x. In each case we've constructed a rational number (p-1/n or p+1/n) that is closer to x than p was. In general, between any two irrational numbers, there's infinitely many rational numbers, and between any two rational numbers, there's (uncountably) infinitely many irrational numbers.