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

Uniqueness of least upper bound, proposition

Let \( E \) be a subset of \( \mathbb{R} \). Then \( E \) can have at most one least upper bound. 

This is a simple proposition.


Proof
Let \( E \) be a subset of \( \mathbb{R} \), and let \( M \) and \( M' \) be two upper bounds of \( E \). If both upper bounds are least upper bounds then both \( M \le M' \) and \( M' \le M \), by the definition of least upper bounds. This implies that \( M = M' \) and that there is only a single least upper bound.

Upper bound def → least upper bound def→ uniqueness of least upper bound → existence of least upper bound → supremum def

Source

Tao, Analysis I