Mills' Theorem/Lemma 2

Lemma for Mills' Theorem
Let:
 * $\N$ denote the set of all natural numbers
 * $\Bbb P$ denote the set of all prime numbers.

Let $p_n$ be the $n$th prime number.

From Difference between Consecutive Primes:
 * $p_{n + 1} - p_n < K {p_n}^{5 / 8}$

where $K$ is an unknown but fixed positive integer.

Let $P_0 > K^8$ be a prime number.

By Lemma 1, there exists an infinite sequence of primes:
 * $P_0, P_1, P_2, \ldots$

such that:
 * $\forall n \in \N_{>0}: {P_n}^3 < P_{n + 1} < \paren {P_n + 1}^3 - 1$

Let us define a mapping $u: \N \to \Bbb P$ as:
 * $\forall n \in \N: \map u n = {P_n}^{3^{-n} }$

Then we have that:
 * $\forall n \in \N_{>0}: \map u {n + 1} > \map u n$