How many convergent subsequences, and how many limits of these subsequences are there in the following sequence:
Justify your answer.
Every number in the interval [0,1] is the limit of a subsequence.
Reason: Let $\displaystyle \alpha\in[0,1]$. For the n-th term of the subsequence, choose the term with denominator n and numerator k such that k/n comes closest to α. Then $\displaystyle |\textstyle\frac kn-\alpha|<1/n.$ Hence the subsequence converges to α.