Consider the usual ultra-power construction of the hyperreals $^*\mathbb R$ with ultrafilter $\mathcal F$. Let $K = [k_n] \in ^*\mathbb N$ be an unlimited hypernatural. My question is does there always exists a (strictly?) monotonically increasing sequence $(a_n)$ such that also $[a_n] = K$. And ...

I've noticed that one question was recently tagged filter. (It is the only question having this tag at the moment.) I was thinking about a few related tags, which might perhaps be useful; but I wanted to ask about the opinion of other users before creating any of these tags. (I might be biased, ...