1. Let $A=\left\{\frac{\sqrt{4}-\sqrt{2}}{\sqrt{8}}, \frac{\sqrt{6}-\sqrt{4}}{\sqrt{24}}, \frac{\sqrt{8}-\sqrt{6}}{\sqrt{48}}, \ldots, \frac{\sqrt{2014}-\sqrt{2012}}{\sqrt{2014 \cdot 2012}}\right\}$.

a) Calculate the sum of the elements of set A.

b) Show that for any subset B of set $A$, the sum of the elements of B cannot be a natural number.
1. a) $S=\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{4}}\right)+\left(\frac{1}{\sqrt{4}}-\frac{1}{\sqrt{6}}\right)+\left(\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{8}}\right)+\cdots+\left(\frac{1}{\sqrt{2012}}-\frac{1}{\sqrt{2014}}\right)=\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2014}}$

b) Since $A$ has all elements strictly positive, the sum of the elements of $B$ will be strictly positive and smaller than the sum of the elements of $A$. If $x$ is the sum of the elements of $B$, it follows that $0<x<\frac{1}{\sqrt{2}}<1$, so $x$ cannot be a natural number.

$3 p$