Let $\mathcal{F}$ be a non-principal ultrafilter on $\omega$. Let $^*\mathbb{N}$ = $\mathbb{N}^\omega/\mathcal{F}$ be an ultrapower. Let $\{n_\alpha\}_{\alpha\in\omega_1}$ be a sequence in $^*\mathbb{N}$ with the following properties:
- It is strictly increasing.
- $\forall \alpha \in \omega_1,\; n_{\alpha+1} = n_\alpha + 1$.
Assuming ZFC, would this sequence be unbounded in $^*\mathbb{N}$, i.e. $\forall n \in {^*\mathbb{N}}\ \exists \alpha \in \omega_1\ n_\alpha > n $? Does it depend on CH?
