proof of divergence of harmonic series (by splitting odd and even terms)

Suppose that the series ${\sum }_{n=1}^{\mathrm{\infty }}1/n$ converged. Since all the terms are positive,we could regroup them as we please, in particular, split the series into two series, that ofeven terms and that of odd terms:

$$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2n}+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2n-1}$$

Since ${\sum }_{n=1}^{\mathrm{\infty }}1/n=2{\sum }_{n=1}^{\mathrm{\infty }}1/(2n)$, we would conclude that

$$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2n}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2n-1}.$$

But , hence , so we would also have

which contradicts the previous conclusion. Thus, the assumption that theseries converged is untenable.