# Additive Group of Integers is Subgroup of Reals

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## 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

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