${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{H(n-1,r)-H(n,r)}& =\frac{(r-1)!(n-r-1)!}{(n-1)!}-\frac{(r-1)!(n-r)!}{n!}\\ \multicolumn{1}{c}{}& =\frac{(r-1)!(n-r-1)!}{(n-1)!}(1-\frac{n-r}{n})\\ \multicolumn{1}{c}{}& =\frac{(r-1)!(n-r-1)!}{(n-1)!}(\frac{r}{n})\\ \multicolumn{1}{c}{}& =\frac{r!(n-r-1)!}{n!}\\ \multicolumn{1}{c}{}& =H(n,r+1)\end{array}}$