Is this proof about why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional correct?

Solution 1:

Here is another proof. Suppose $\mathbb R$ is a finite-dimensional $\mathbb Q$ vector space, with basis $\{x_1,\dots,x_n\}$. But $span\{x_1,\dots,x_n\}\cong\mathbb Q^n$ has cardinality $\aleph_0$ while $\mathbb R$ has cardinality $2^{\aleph_0}$, which is absurd.

Solution 2:

Your proof is sufficient for the question that was asked once you show that those vectors are independent. It could well be that you have already shown that, perhaps as a lead-in for this question. As Kenta S has shown, you can prove more, that the dimension is $\mathfrak c$, but you were not asked for that.