Infimum of Bounded Below 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 below subset of $\R$ such that $H \ne \O$.

Let $l = \map \inf H$ be the infimum of $H$.


Then:

$l \in \map \cl H$

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


Proof

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

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

From Distance from Subset of Real Numbers:

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

Thus by definition of distance from subset:

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

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

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

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

The result follows from Condition for Point being in Closure.

$\blacksquare$


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