$\displaystyle 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{1}{2n-1}-\frac{1}{2n}=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\f rac{1}{2n}$
Consider $\displaystyle k_n = 1 + \frac{1} {2} + \cdots + \frac{1} {n}.$ Note that $\displaystyle \frac{1} {2}k_n = \frac{1} {2} + \frac{1} {4} + \cdots + \frac{1} {{2n}}.$ The LHS is $\displaystyle k_{2n} - 2\left( {\frac{1} {2}k_n } \right)$ & the RHS is $\displaystyle k_{2n}-k_n,$ as required $\displaystyle \blacksquare$