Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as 
\begin{equation}
J(a,b) = 1 - \frac{||a\odot b||_1}{||a\odot b||_1+||a-b||_1}
\end{equation}
where $\odot$ is the Hadamard product (element-wise product).

Is it a metric? Note that $a,b \in [0,1]^p$ rather than $\{0,1\}^p$.

I've tried with the naive approach. After some messy algebra, I need to prove the following 
\begin{equation}
||a-b||_1||a\odot c||_1||b\odot c||_1 + ||a-b||_1||b-c||_1||a\odot c||_1 + ||a-b||_1 ||b-c||_1 ||a-c||_1 + ||b-c||_1||a\odot b||_1 ||a\odot c||_1 + ||a-b||_1||b-c||_1||a\odot c||_1 \geq ||a-c||_1||a\odot b||_1 ||b\odot c||_1.
\end{equation}