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 strictly increasing sequence in $^*\mathbb{N}$.

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?