Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for him?

The algebraist argues that the real numbers are a silly construction because any real number can be approximated to arbitrarily high precision by the rational numbers - i.e., given any real number $r$ and any $\epsilon>0$, the set $\left\{x\in\mathbb{Q}:\left|x-r\right|<\epsilon\right\}$ is nonempty, thus sating the mad gods of algebra.

As @J.M. and @75064 pointed out to me in chat, we do start having some topology problems, for example that $f(x)=x^2$ and $g(x)=2$ are nonintersecting functions in $\mathbb{Q}$. They do, however, come *arbitrarily close* to intersecting, i.e. given any $\epsilon>0$ there exist rational solutions to $\left|2-x^2\right|<\epsilon$. The algebraist doesn't find this totally unsatisfying.

Where is this guy

reallygoing to start running into trouble? Are there definitions in analysis which simply can't be reasonably formulated without leaving the rational numbers? Which concepts would be particularly difficult to understand without the rest of the reals?