Putnam 1946 B4

For each positive integer $n$ , put $$p_n =\left(1+\frac{1}{n}\right)^{n}, P_n =\left(1+\frac{1}{n}\right)^{n+1}, h_n = \frac{2 p_n P_{n}}{ p_n + P_n }.$$ Prove that $h_1 < h_2 < h_3 <\ldots$