Can someone please tell me where this line of reasoning goes wrong?
False proof for the convergence of the alternating harmonic series:
Break the series $S = 1 - 1/2 + 1/3 - 1/4 + \dots$ into the following "subseries":
$S_1=1 - 1/2 - 1/4 - 1/8 - \dots$
$S_3=1/3 - 1/6 - 1/12 - 1/24 - \dots$
$S_5=1/5 - 1/10 - 1/20 - 1/40 - \dots$
$S_7=1/7 - 1/14 - 1/28 - 1/56 - \dots$
etc.
First, it seems that no term from the original series occurs in more than one subseries. If a term did occur in more than one, then we would have $z \cdot 2^i= y \cdot 2^j$, where $z$ and $y$ are odd. So $z \cdot 2^{i-j}=y$, and the only way this can happen is if $i=j$ and $z=y$.
Second, it seems that every term from the original series can be found in one of the subseries. Take any integer $k$ and decompose it into $k=(2^i) \cdot z$, where $z$ is odd. Then the $k$th term of the original series can be found in the $i+1$ term of $S_z$.
Therefore $S=S_1 + S_3 + S_5 + S_7 + \dots$, and all the $S_i=0$, so $S=0$.