Supremum of Bounded Above Set of Reals is in Closure

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Theorem

Let $\R$ be the real number line under the Euclidean metric.

Let $H \subseteq \R$ be a bounded above subset of $\R$ such that $H \ne \varnothing$.

Let $u = \map \sup H$ be the supremum of $H$.


Then:

$u \in \map \cl H$

where $\map \cl H$ denotes the closure of $H$ in $\R$.


Proof

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.

Let $\map {B_\epsilon} u$ be the open $\epsilon$-ball of $u$ in $\R$.

From Distance from Subset of Real Numbers:

$\map d {u, H} = 0$

Thus by definition of distance from subset:

$\exists x \in H: \map d {u, x} < \epsilon$

Thus $x \in \map {B_\epsilon} u$.

As $x \in H$ and $x \in \map {B_\epsilon} u$, from the definition of intersection:

$x \in H \cap \map {B_\epsilon} u$

The result follows from Condition for Point being in Closure.

$\blacksquare$


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