The $2013$ numbers $$\frac{1}{1\times 2}, \frac{1}{2\times 3},\frac{1}{3\times 4},...,\frac{1}{2013 \times 2014}$$are arranged randomly on a circle. (a) Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{4000}$ . (b) Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{10000}$ .