TITLE: Example of a Countable set which has volume zero.
QUESTION [6 upvotes]: 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...

REPLY [3 votes]: 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 $<\epsilon$.