Additive Group of Integers is Subgroup of Reals

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 Normal Subgroup of Reals, $\struct {\Q, +}$ is a subgroup of $\struct {\R, +}$.

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

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.2$. Subgroups: Example $91$