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 the $*\mathbb{N}$ with the following properties:

1. It is strictly increasing.
2. $\forall \alpha \in \omega_1\ n_{\alpha+1} = n_\alpha + 1$.

Assuming ZFC, would this sequence be unbounded in the $*\mathbb{N}$, i.e. $\forall n \in *\mathbb{N}\ \exists \alpha \in \omega_1\ n_\alpha > n $? Does it depend on CH?