$\mathbb{R}$ is not a direct sum of its subgroups

Question

How to prove the set of real numbers under addition; i.e., $(\mathbb{R}, +)$, is not the direct sum of two of its proper subgroups?

Answer 1

For me, your statement is false: $\mathbb{R}$ is a $\mathbb{Q}$-vector space of infinite dimension, so two proper subspaces $A,B$ may be found so that $\mathbb{R}= A \oplus B$. In particular, it gives a decomposition of $\mathbb{R}$ as an additive group.