Subgroup/Examples/Natural Numbers in Multiplicative Group of Real Numbers

Example of Closed Subset which is not a Subgroup
Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.

Consider the algebraic structure $\struct {\N_{> 0}, \times}$ formed by the non-zero natural numbers under multiplication.

Then $\struct {\N_{> 0}, \times}$ is not a subgroup of $\struct {\R_{\ne 0}, \times}$.

Proof
From Non-Zero Natural Numbers under Multiplication form Commutative Monoid, $\struct {\N_{> 0}, \times}$ is a monoid whose identity is $1$.

But there exists no $x \in \N_{> 0}$ such that, for example, $x \times 2 = 1$.

So $\N_{> 0}$ is not a subgroup of $\struct {\R_{\ne 0}, \times}$.