I am reading Stability Conditions on Triangulated Categories Prop. 8.1. When Bridgeland proves the continuity of $\log \frac{m_{\sigma_2}(E)}{m_{\sigma_1}(E)}$ he says:
But now one can choose $\phi$ so that $$\sum_k m_\sigma(\beta_k(E))\leq\left(1+\frac2n\right)m_\sigma(E)$$
Why is this true? I think that the $\left(1+\dfrac2n\right)$ factor could be the ratio between the length of the two intervals $I_k$ and $J_k$ defined a few lines before, but why is it related? Also, why is it related to the choice of $\phi$?
Thank you.