2 Bertrand’s postulate

Theorem 2.1

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For any positive natural number, there is a prime which is greater than it, but no more than twice as large.

Proof ▶

TODO: make this follow the book proof more closely!

2.1 Appendix: Some estimates

Theorem 2.2

For all \(n \in \mathbb {N}\)

\[ \log n + \frac1n {\lt} H_n {\lt} \log n + 1. \]

Proof ▶

TODO

Theorem 2.3

For all \(n \in \mathbb {N}\)

\[ n! = n(n -1)! {\lt} ne^{n \log n - n + 1}= e\left(\frac ne\right)^n. \]

Proof ▶

TODO

Theorem 2.4

\[ \binom {n}{k} \le \frac{n^k}{k!} \le \frac{n^k}{2^{k - 1}} \]

Proof ▶

TODO