Let $V$ be the set of real numbers. Regard $V$ as a vector space over the field of rational numbers, with the usual operations. Prove that this vector space is not finite-dimensional.
MY ATTEMPT
Suppose $V$ is finite dimensional. Thus it is spanned by some set $S = \{r_{1},r_{2},\ldots,r_{n}\}$ of linearly independent real numbers. Therefore any real number $R$ can be written as \begin{align*} R = q_{1}r_{1} + q_{2}r_{2} + \ldots q_{n}r_{n} \end{align*}
where $q_{i}$ are rationals. From there, we conclude that $V$ is countable, once $|V| = |\textbf{Q}^{n}| = |\textbf{Q}|$, which is a contradiction.
Am I on the right track? If so, is there another approach to this problem? Thanks in advance!
