NO-ABEL 1995

Show that there exists a sequence $x_1,x_2,...$ of natural numbers in which every natural number occurs exactly once, such that the sums $\sum_{i=1}^n \frac{1}{x_i}$ , $n = 1,2,3,...$ , include all natural numbers.

Let $x_i,y_i$ be positive real numbers, $i = 1,2,...,n$ . Prove that $$\left( \sum_{i=1}^n (x_i +y_i)^2\right)\left( \sum_{i=1}^n\frac{1}{x_iy_i}\right)\ge 4n^2$$