Find the limit superior of an enumeration of an interval in the rational numbers

Question

How would I go about finding lim sup $a_n$, where $\{a_n\}$ is an enumeration of the rationals in the interval $[0,1]$? Honestly, I don't even know where to start.

Answer 1

For any $\varepsilon>0$,the numbers $1-\dfrac1n$ such that $1-\varepsilon<1-\dfrac1n<1$ are an infinity, hence for any enumeration of the rationals in $[0,1]$, an infinity of them are in the enumeration with an index as large as we please. Thus $\limsup a_n=1$.