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Math and science::Analysis::Tao::05. The real numbers

Existence of least upper bound, theorem

Let E be a subset of R. If E has some upper bound, then it must have a least upper bound.

While 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, (an)n=1, consists of only upper bounds, and the other, (bn)n=1 only of not upper bounds. There is a relationship between elements in the sequences: ak and bk differ by a rational 1n. As n increases, it is shown that both sequences are equivalent and define the same real, S. Then, considering any other upper bound M, it must be greater or equal to all elements of (bn)n=1, otherwise it would not be an upper bound. Consequently, it must be greater to or equal to S. This makes S 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


Source

Tao, Analysis I