Characterizing Property of Infimum of Subset of Real Numbers

Theorem
Let $S \subset \R$ be a nonempty subset of the real numbers.

Let $S$ be bounded below.

Let $\alpha \in \R$.


 * 1) $\alpha$ is the infimum of $S$
 * 2) * $\alpha$ is a lower bound for $S$ and
 * 3) *$\forall \epsilon>0$ there exists $x \in S$ with $x < \alpha + \epsilon$
 * 1) *$\forall \epsilon>0$ there exists $x \in S$ with $x < \alpha + \epsilon$

1 implies 2
Let $\alpha$ be the infimum of $S$.

Then by definition, $\alpha$ is a lower bound for $S$.

Let $\epsilon>0$.

Because $\alpha+\epsilon>\alpha$, it is not a lower bound for $S$.

Thus there exists $x\in S$ with $x < \alpha + \epsilon$.

2 implies 1
Let $\alpha$ be a lower bound of $S$ such that $\forall \epsilon>0$ there exists $x \in S$ with $x < \alpha + \epsilon$.

Let $d\in \R$ be a lower bound of $S$.

We have to prove that $d \leq \alpha$.

Suppose $d > \alpha$.

Let $\epsilon = d-\alpha>0$.

Then there exists $x\in S$ such that $x < \alpha + (d-\alpha) = d$.

But then $d$ is not a lower bound of $S$.

Also see

 * Characterizing Property of Supremum of Subset of Real Numbers