Number of distinct limits of subsequences of a sequence is finite? "The number of distinct limits of subsequences of a sequence is finite?"
I've been mulling over this question for a while, and I think it is true, but I can't see how I might prove this formally. Any ideas? 
Thanks

The answer is no.
Think for instance about subsequences of $(cos(n))$: after showing that the integers are dense in $[0,2\pi]$, it follows that for every real number $k$ in $[-1,1]$ there exists a subsequence $(cos(n_k))_{k \in \mathbb{N}}$ which converges to $k$. In particular, not only the number of distinct limits of subsequences is infinite, but uncountably infinite.
