Prove $\exists$ infinite set $S \subset \mathbb{R}$ s.t. $\forall x \in \mathbb{R}$, $x$ can be represented as a unique linear combination of numbers in $S$ with rational coefficients.
I'm not asking if every vector space has a basis. I already have that proven in my notes so this is NOT a duplicate.
I have that $\mathbb{R}$ as a vector space over $\mathbb{Q}$ has a basis and therefore by definition of a basis, $\mathbb{R}$ can be represented as a linear combination of some vectors $x_i \in B_i$ multiplied by coefficients in $\mathbb{Q}$, but I don't have that this representation is unique.
Since every vector space has a basis, we can let $S$ be a Hamel basis of $\mathbb{R}$ as $\mathbb{Q}$-vector space. Then every real $x$ can be represented as a unique linear combination of elements of $S$ with rational coefficients.