Help with following equality Could you please help me understand how this is?
$$\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{2n-1}-\frac{1}{2n}=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n} \,.$$
Thank you.

$\text{L.H.S.}$
$=\displaystyle \frac 1 1-\frac 1 2+\frac 1 3 - \frac 1 4 +\cdots+\frac 1 {2n-1} - \frac 1 {2n}$
$\displaystyle=\frac 1 1+\frac 1 2+\frac 1 3 + \frac 1 4 +\cdots+\frac 1 {2n-1} + \frac 1 {2n}-2\Big(\frac 1 2+\frac 1 4+\cdots+\frac 1 {2n}\Big)$
$\displaystyle=\frac 1 1+\frac 1 2+\frac 1 3 + \frac 1 4 +\cdots+\frac 1 {2n-1} + \frac 1 {2n}-\Big(\frac 1 1+\frac 1 2+\cdots+\frac 1 {n}\Big)$
$=\displaystyle\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n} $
$= \text{R.H.S.}$
Q.E.D.
