I know that two arithmetic sequences of real numbers that both contain the number $0$ have terms that are arbitrarily close together, that is, for any $\varepsilon>0$ there exists a real number such both sequences have a term within $\varepsilon$ of that number but $0$ isn't within $\varepsilon$ of that number. Is this necessarily true for three (or more) arithmetic sequences that all contain the number $0?$ If not, for what number of arithmetic sequences does this fail?
My motivation for this question was that I wanted to devise a scale in music that had very close approximations to nice intervals like the octave, perfect fifth, major third, harmonic seventh, and other nice intervals relative to the number of notes per octave, but I wanted to know whether we can get arbitrarily close before attempting an "optimal" such scale.
Yes, this is always true; in fact, there is not just a real number that every sequence gets within $\epsilon$ of but an integer. Let me reformulate the question a bit first. Suppose you have a lattice $A\subset\mathbb{R}^n$ (e.g., the lattice consisting of points whose coordinates are from each of $n$ chosen arithmetic sequences) and a nonzero vector $v\in\mathbb{R}^n$ (e.g., $v=(1,1,\dots,1)$). Then I claim that there are nonzero integer multiples of $v$ that get arbitrarily close to points of $A$.
To prove this, note that by compactness of the quotient group $\mathbb{R}/A$, the sequence $(nv)_{n\in\mathbb{N}}$ has a subsequence that converges mod $A$. So, for any $\epsilon>0$, there exist distinct $m,n\in\mathbb{N}$ such that $mv$ is within $\epsilon$ of $nv$ mod $A$. Then $(m-n)v$ is a nonzero integer multiple of $v$ that is within $\epsilon$ of a point of $A$.