Let $\mathbb{N}_0$ be the usual structure of natural numbers (including 0) in the usual signature with $\leq , +, \times, 0$ and 1. Let $U$ be a nonprincipal ultrafilter on $\omega$, and let $\textbf{N}$ be the ultrapower $\mathbb{N}_0^{\omega}/U$. We will identify $\mathbb{N}_0$ with its image in $\mathbb{N}_0^{\omega}/U$ under the natural (diagonal) embedding, and refer to the elements of $\mathbb{N}_0$ as standard $natural \ numbers$. The definition of a prime number remains standard: $x$ is prime if $\forall y:x = y \times z \implies (y = 1 \vee z = 1)$. Prove:
(1) That in $\textbf{N}$ there exists non-standard primes: primes that are greater than any standard prime.
(2) That any elements of $\textbf{N}$ is smaller than some (non-standard) prime.
I have been working on this problem for a long time, any help is appreciated!