I've been studying Spivak's "Calculus on Manifolds" and I'm thinking about the following question:
Let $C \subset [0,1]^2$ be the union of all $\{p/q\} \times [0,1/q]$, where $p/q$ is a rational number in [0,1] written in lowest terms. Show $C$ has content $0$.
What I've tried to show is that $C$ has measure $0$, since it is the countable union of sets of measure $0$, hence has measure $0$. Then I tried to show $C$ is compact, because that would imply it has content $0$, but the problem is that it isn't compact, since it's not closed; $C$ contains points of the form $(p/q,0)$ and these can be used to approximate $(\text{irrational number},0)$ arbitrarily well, but the latter clearly doesn't belong to $C$, so I'm kinda stuck. I want to show it has content $0$ probably using the definition, that is finding a finite cover of $C$ by means of closed rectangles with arbitrarily small area, but I'm not sure how to construct it. Any hints will be greatly appreciated.
NOTE: I'm unsure if this is actually related to Lebesgue theory, since Spviak doesn't mention it. Feel free to change the tags.
HINT: Can you show that, given any $\epsilon>0$, you can cover your set by finitely many rectangles in $[0,1]^2$ with total area $<\epsilon$?