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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

Tao, Analysis I