Example of a Countable set which has volume zero. The following question was asked in my exam which states:  

Give an example of a countable set which has volume zero.
Hint: show that if $S=\{a_1,a_2,\ldots,a_n,\ldots\}$ is the set of points which form a convergent sequence then S has volume zero.    

I still can't understand what's the use of a convergent sequence in showing that $S$ has volume zero. Can anyone please help me understanding this...
 A: Given a convergent sequence $a_n\to a$, take a rectangle $R$ centered at $a$ of volume $\frac12\epsilon$. There are only finitely many $n$ with $a_n\notin R$. For each of these pick a rectangle around $a_n$ of volume $2^{-n}\epsilon$. Then the total volume of these finitely many rectangles is $<\epsilon$.
A: Unless I misunderstand, there is a much simpler solution. Consider the set $S = \{(r, 0) \mid r \in \mathbb{Q} \cap [0, 1]\}$. Then we look at rectangles of the form (roughly, you should tweak this accordingly) $R = [0, 1] \times [-\epsilon, \epsilon]$. These contain $S$, and their volume is going to be $2\epsilon$.
A: To apply what the definition in your notes should be, for each $n$, choose a rectangle of side $\varepsilon^{1/d}/2^n$ which contains $a_n$, then the union of these rectangles contains $S$ and its volume is at most the sum of the volumes $(\varepsilon^{1/d}/2^n)^d$ of the rectangles, which is less that $\varepsilon$, QED.
The suggestion in your notes to include $S$ in a finite union of rectangles is inaccurate in general (consider, for example, $S=\mathbb Z^d$). In the particular case when $S$ is the set of points of a converging sequence, one can proceed as follows.
By definition of convergence, for every $\varepsilon$, there exists $N$ such that every $a_n$ is at distance at most $\varepsilon$ from the limit. Thus all the points $a_n$ with $n\geqslant N$ are in a rectangle of side $2\varepsilon$. Use rectangles of side $\varepsilon/2^n$ for the $N$ first points. Then the total volume of the rectangles is at most $\varepsilon^d+(2\varepsilon)^d$, QED.
