114 THE ARITHMETIC AND GEOMETRIC MEANS. THE ARITHMETIC AND GEOMETRIC MEANS HARALD BOHR*. Let n be a positive integer and a v ..., a lt positive numbers; then the famous theorem of the arithmetic and geometric means states that n i.e. (#!+...+flj Of this theorem a number of different proofs are known [cf. Hardy, Littlewood, and Polya, Inequalities (Cambridge, 1934)]. In this note I shall give a new one. Even if this proof is more curious than simple, and moreover does not show that the sign of equality holds only in the case a x = ... = a. n , it may perhaps be worth while to indicate one more way of arriving at the classical inequality in question. It is trivial that since the product on the right side is just one of the terms of the polynomial development of the left side. We have to prove that the factor n\ may be replaced by n n . This can be done by help of the following artifice. Let q > 1 be another integer. From the polynomial development we have immediately and hence {qn) n Now let q-> oo. Then (qn) n / q -+1, and we get the desired result. Maglevaenet 9, Charlottenlund, Denmark. * Received 12 February, 1935; read 14 February, 1935. f In fact the numerator (qn)! is evidently greater than [(q— 1)! n?- 1 }", since each of then products k(k+n)...{k+(q— l)n} [k=l, ..., n] is greater than l.n.2n...(q—\)n={q—l)\n'i-\