summation - Easy question regarding this proof

I do not understand a small step in a proof I'm reading at the moment. Why are the following things equal?

$$\sum_{k=1}^{n} \frac{1}{2k-1} - \frac{1}{2} \sum_{k=1}^{n} \frac{1}{k} = \sum_{k=1}^{2n} \frac{1}{k} - \sum_{k=1}^{n} \frac{1}{k}$$

Answer

Separating out the odd & the even terms in the denominator,

$$\sum_{k=1}^{2n}\frac1k=\sum_{k=1}^n\left(\frac1{2k-1}+\sum_{k=1}^n\frac1{2k}\right)$$

$$=\sum_{k=1}^n\frac1{2k-1}+\sum_{k=1}^n\frac1{2k}$$

$$=\sum_{k=1}^n\frac1{2k-1}+\frac12\sum_{k=1}^n\frac1k$$