\begin{align*}
    \frac{3(\lfloor \log{n} \rfloor + 1)}{W} \sum_{i=1}^{n-1}{2 i  v_i  w_i} \geq & \ \frac{\lfloor \log{n} \rfloor + 1}{W} \sum_{i=1}^{n}{2 i  v_i  w_i} \\
    \geq & \ \frac{1}{W} \sum_{i=1}^{n}{2 ((\lfloor \log{n} \rfloor + 1)  i - 1)  v_i  w_i} \\
    \geq & \ \frac{1}{W} \sum_{i=1}^{n}{2 ((\lfloor \log{n} \rfloor + 1)  2^{\lfloor \log{i} \rfloor} - 1)  v_i  w_i}.
 \end{align*}
