The definition of a harmonic series is that for any $H_{k-1},H_k,H_{k+1} \in \{H_1,H_2, ... H_n\}$, $$\frac{2}{H_k} = \frac{1}{H_{k-1}} + \frac{1}{H_{k+1}}$$
If any $H_k$ is zero, then $\frac 1 {H_k}$ is not defined, hence zero cannot be an element of a harmonic progression.
Does this proof hold good? If not, give an example of a harmonic series with zero as one of it's elements.
Technically, this is debatable.
We can easily state
$$H_{k-1}\ne0\land H_{k+1}\ne0\implies H_{k}\ne0$$
but it could turn out that all $H$ are zero and the "definition" can never be used. A zero sequence cannot be disproven harmonic.