Hello all, (*) I have a question about convergent subsequences. Specifically I am looking for an example of a sequence that is unbounded but who has convergent subsequences in the interval [0,1]. A similar question would be to have an unbounded sequence, but who has a convergent subsequence to a specific number, let's say 0. For this I would take the sequence: a_n = -1,0,1,-2,0,2,-3,0,3,...,-n,0,n Can I do a similar thing for (*)? i.e Can I take the following sequence: a_n = -1, [0,1], 1, -2, [0,1], 2,...,-n, [0,1], n My intuition tell me no since there will be infinite terms in each of the intervals [0,1]. Thanks in advance for any and all help!