Infimum is not necessarily Smallest Element/Proof

Proof
Let $V$ be the subset of the real numbers $\R$ defined as:
 * $V := \set {x \in \R: x > 0}$

From Infimum of Subset of Real Numbers: Example 3, $V$ admits an infimum:


 * $\inf V = 0$

But $V$ has no smallest element, as follows.

We note that $\inf V = 0 \notin V$.

$x \in V$ is the smallest element of $V$.

Then $\dfrac x 2$ is also in $V$ as $\dfrac x 2 > 0$.

But $\dfrac x 2 < x$ which contradicts $x$ as being the smallest element of $V$.

Hence there can be no smallest element of $V$.