Let $S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\cdots$.
Or $S=(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots)+(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\cdots)$.
$\implies S=(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots)+\frac{1}{2}({1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots)$.
$\implies S=(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots)+\frac{1}{2}(S)$.
$\implies \frac{S}2=1+(\frac{1}{3}+[\frac{1}4-\frac{1}4]+\frac{1}{5}+[\frac{1}6-\frac{1}6]+\frac{1}{7}+[\frac{1}8-\frac{1}8]+\cdots)$.
$\implies \frac{S}2=1+(\frac{1}{3}-\frac{1}4+\frac{1}{5}-\frac{1}6+\frac{1}{7}-\frac{1}8+\cdots)+\frac{1}2(\frac{1}2+\frac{1}{3}+\frac{1}{4}+\cdots)$.
$\implies \frac{S}2=1+(\frac{1}{3.4}+\frac{1}{5.6}+\frac{1}{7.8}+\cdots)+\frac{1}2(S-1)$.
$\implies 0=\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+\frac{1}{7.8}+\cdots$.
Which we know is not correct.
Also, we know that we do face this type of discrepancies when we are dealing with divergent series(as is the case when we put $x=1$ in the infinite geometric series of $\frac{1}{1-x}$) in the usual manner of dealing with convergent series.
Therefore, $S$ must be a divergent series.
Can we conclude something more/else from the last equation?
EDIT
Can we conclude that since the series is divergent and the partial sum of the terms is always increasing so, the only way it can be divergent is it($S$) has to be $\infty$?
Yes, we can prove the divergence of the harmonic series like that, but one must at the beginning state something along the lines of "suppose the harmonic series is convergent".
Once that assumption is introduced, it is a valid proof by contradiction. Since all terms in the harmonic series are positive, if it were convergent, it would be absolutely convergent, hence also all of its subseries would be absolutely convergent, and all manipulations you made are valid for absolutely convergent series.
Since at the end you reach a manifestly false assertion by means that are valid if the assumption is true, it correctly follows that the assumption must have been false, i.e. that the harmonic series diverges.