Consider the set of rational numbers {1, 1.4, 1.41, 1.414, 1.4142, ...} converging to the square root of 2. If all we knew were rational numbers, this set would have no supremum. If we allow real numbers, there is a unique supremem.Back
Examples 2.4.3(b): |
Consider the set of rational numbers {1, 1.4, 1.41, 1.414, 1.4142, ...} converging to the square root of 2. If all we knew were rational numbers, this set would have no supremum. If we allow real numbers, there is a unique supremem. |
is the least upper bound (although each of these numbers is an upper bound),
because if x was that least upper bound, then we can find a rational
number between and
x. That rational number would then be an upper bound smaller than
x, which is a contradiction.
is the least upper bound, because if x was that least upper bound,
there is some element of the set between x and
. But then x is not
an upper bound, which is a contradiction.
If we consider this set as a subset of the real numbers, then the least
upper bound of this set is .