# Thread: Subsequences

1. ## Subsequences

How many convergent subsequences, and how many limits of these subsequences are there in the following sequence:

Justify your answer.

2. Originally Posted by matty888
How many convergent subsequences, and how many limits of these subsequences are there in the following sequence:

Justify your answer.
An infinite amount...

1/2 + 1/3 + 1/4 + 1/5 ....

1/3 + 1/4 + 1/5 + 1/6 .....

3. ## Thanks(???)

Thank you, could I justify this by writing in the form 1/n+1 and getting the sum to infinity??(Or something like that)

4. Originally Posted by matty888
Thank you, could I justify this by writing in the form 1/n+1 and getting the sum to infinity??(Or something like that)
Yes, with different starting points.

5. Originally Posted by matty888
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.