Upper Bounds for Prime Numbers

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Let $p: \N \to \N$ be the prime enumeration function.

Then $\forall n \in \N$, the value of $p \left({n}\right)$ is bounded above.

In particular:

Result 1

$\forall n \in \N: p \left({n}\right) \le 2^{2^{n-1}}$

Result 2

$\forall n \in \N: p \left({n}\right) \le \left({p \left({n - 1}\right)}\right)^{n-1} + 1$

Result 3

$\forall n \in \N_{>1}: p \left({n}\right) < 2^n$


In the first two cases it can be seen that the limit found is wildly extravagantly large. However, they are results that are easily established, and they have their uses.