CMO 1973 Problem 7

Observe that $$\frac{1}{1}= \frac{1}{2}+\frac{1}{2}; \frac{1}{2}=\frac{1}{3}+\frac{1}{6}; \frac{1}{3}=\frac{1}{4}+\frac{1}{12}; \frac{1}{4}= \frac{1}{5}+\frac{1}{20}.$$ State a general law suggested by these examples, and prove it.

Prove that for any integer $n$ greater than 1 there exist positive integers $i$ and $j$ such that $$\frac{1}{n}= \frac{1}{i(i+1)}+\frac{1}{(i+1)(i+2)}+\frac{1}{(i+2)(i+3)}+\cdots+\frac{1}{j(j+1)}.$$