Additive Group of Integers is Subgroup of Reals

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\R, +}$ be the additive group of real numbers.


Then $\struct {\Z, +}$ is a subgroup of $\struct {\R, +}$.


Proof

From Additive Group of Integers is Subgroup of Rationals, $\struct {\Z, +}$ is a subgroup of $\struct {\Q, +}$.

From Additive Group of Rationals is Subgroup of Reals, $\struct {\Q, +}$ is a subgroup of $\struct {\R, +}$.

Thus $\struct {\Z, +}$ is a subgroup of $\struct {\R, +}$.

$\blacksquare$


Sources