Why is $\mathbf{Q}$ dense in $\mathbf{R}$?
Does this imply that (Edit: the field of algebraic numbers) $\overline{\mathbf{Q}}$ is dense in $\mathbf{C}$?
For the first question, I guess one has to show that for every $x$ in $\mathbf{R}$ and every $\epsilon >0$, there exists a rational number $q$ such that $\vert x- q \vert < \epsilon$. Does the proof of this rely upon methods from Diophantine Approximation or is it easier than this?