Math and science::Analysis::Tao::05. The real numbers
Existence of least upper bound, theorem
LetWhile it may seem banal at first, the result is quite amazing. To appreciate this, consider why a set of rationals does not necessarily have a rational least upper bound.
Proof outline
The proof of this is somewhat involved (but not difficult). The proof (p118) is a great example of the utilization of the properties of Cauchy sequences and equivalent sequences.
It starts by constructing two Cauchy sequences. One sequence, , consists of only upper bounds, and the other, only of not upper bounds. There is a relationship between elements in the sequences: and differ by a rational . As increases, it is shown that both sequences are equivalent and define the same real, . Then, considering any other upper bound , it must be greater or equal to all elements of , otherwise it would not be an upper bound. Consequently, it must be greater to or equal to . This makes less than or equal to all other upper bounds.
Example
Upper bound def → least upper bound def→ uniqueness of least upper bound → existence of least upper bound → supremum def