This is a homework problem.
If $(x_n)$ is a bounded sequence of real numbers, prove that there exists a subsequence of $(x_n)$ that converges towards the standard part of the hyperreal $[(x_n)]$, ie. the equivalence class of $(x_n)$.
Now a standard part of a hyperreal $[(x_n)]$ is a number $\alpha$ such that $ \forall r>0, \ \ |(x_n-\alpha)|
Now if a subsequence $(x_k)$ converges to $\alpha$, then $\forall \epsilon>0 \ \ \exists N_{\epsilon} \in \mathbb{N}, \ \forall k>N_{\epsilon},\ \ |x_k-\alpha|<\epsilon $
Now I would need to use the fact that $ \forall r>0, \ \ |(x_n-\alpha)|
The problem I have is basically that it's been far too long since I last needed this kind of proper math, sequences and so, have been to simulation and probability theory and such for too long.