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 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 $91$