Mills' Theorem/Lemma 1

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

Then we have that:
 * $\forall N > K^8 \in \Z: \exists p \in \Bbb P: N^3 < p < \paren {N + 1}^3 - 1$

Proof
Let $p_n$ be the greatest prime less than $N^3$.

Therefore:
 * $N^3 < p_{n + 1} < \paren {N + 1}^3 - 1$