Pocklington’s Theorem and Bertrand’s Postulate

The following propositions are true: (1) For all real numbers r, s such that 0 ≤ r and s · s < r · r holds s < r. (2) For all real numbers r, s such that 1 < r and r · r ≤ s holds r < s. (3) For all natural numbers a, n such that a > 1 holds an > n. (4) For all natural numbers n, k, m such that k ≤ n and m = b n2 c holds (n m ) ≥ (n k ) . (5) For all natural numbers n, m such that m = b n2 c and n ≥ 2 holds (n m ) ≥ 2n n . (6) For every natural number n holds (2·n n ) ≥ 4n 2·n . (7) For all natural numbers n, p such that p > 0 and n | p and n 6= 1 and n 6= p holds 1 < n and n < p.