$$
\begin{array}{l} \sum_ {k = 0} ^ {K (\mu_ {0}, a, \delta , \varepsilon_ {\mathrm {d i s t}})} \frac {2 (\mu_ {k} (a ^ {2} + 1) + 3 a ^ {2}) (g _ {\mu_ {k + 1}} (W _ {\mu_ {k} , \epsilon_ {k}}) - g _ {\mu_ {k + 1}} (W _ {\mu_ {k + 1} , \epsilon_ {k + 1}}))}{\epsilon_ {k} ^ {2}} \\ \leq 2(\mu_{0}(a^{2} + 1) + 3a^{2})\left(\sum_{k = 0}^{\widehat{K} (\mu_{0},a,\delta ,\varepsilon_{\mathrm{dist}}) - 1}\frac{(g_{\mu_{k + 1}}(W_{\mu_{k},\epsilon_{k}}) - g_{\mu_{k + 1}}(W_{\mu_{k + 1},\epsilon_{k + 1}}))}{\epsilon_{k}^{2}} +\sum_{k = \widehat{K} (\mu_{0},a,\delta ,\varepsilon_{\mathrm{dist}})}^{K(\mu_{0},a,\delta ,\varepsilon_{\mathrm{dist}})}\frac{(g_{\mu_{k + 1}}(W_{\mu_{k},\epsilon_{k}}) - g_{\mu_{k + 1}}(W_{\mu_{k + 1},\epsilon_{k + 1}}))}{\epsilon_{k}^{2}}\right) \\ = 2\left(\mu_{0}\left(a^{2} + 1\right) + 3a^{2}\right)\left(\sum_{k = 0}^{\widehat{K} (\mu_{0},a,\delta ,\varepsilon_{\text{dist}}) - 1}\frac{\left(g_{\mu_{k + 1}}\left(W_{\mu_{k},\epsilon_{k}}\right) - g_{\mu_{k + 1}}\left(W_{\mu_{k + 1},\epsilon_{k + 1}}\right)\right)}{\beta^{2}a^{2}\mu_{k}^{2}} + \sum_{k = \widehat{K} (\mu_{0},a,\delta ,\varepsilon_{\text{dist}})}^{K(\mu_{0},a,\delta ,\varepsilon_{\text{dist}})}\frac{\left(g_{\mu_{k + 1}}\left(W_{\mu_{k},\epsilon_{k}}\right) - g_{\mu_{k + 1}}\left(W_{\mu_{k + 1},\epsilon_{k + 1}}\right)\right)}{\mu_{k}^{3}}\right) \\ \leq 2(\mu_{0}(a^{2} + 1) + 3a^{2})\left(\sum_{k = 0}^{\widehat{K} (\mu_{0},a,\delta ,\varepsilon_{\text{dist}})^{-1}}\frac{(g_{\mu_{k + 1}}(W_{\mu_{k}},\epsilon_{k}) - g_{\mu_{k + 1}}(W_{\mu_{k + 1},\epsilon_{k + 1}}))}{\beta^{6}a^{6}} + \sum_{k = \widehat{K} (\mu_{0},a,\delta ,\varepsilon_{\text{dist}})}^{K(\mu_{0},a,\delta ,\varepsilon_{\text{dist}})}\frac{(g_{\mu_{k + 1}}(W_{\mu_{k}},\epsilon_{k}) - g_{\mu_{k + 1}}(W_{\mu_{k + 1},\epsilon_{k + 1}}))}{\mu_{k}^{3}}\right) \\ \leq 2(\mu_{0}(a^{2} + 1) + 3a^{2})\left( \sum_{k = 0}^{K(\mu_{0},a,\delta ,\varepsilon_{\text{dist}})}\frac{(g_{\mu_{k + 1}}(W_{\mu_{k},\epsilon_{k}}) - g_{\mu_{k + 1}}(W_{\mu_{k + 1},\epsilon_{k + 1}}))}{\beta^{6}a^{6}} + \sum_{k = 0}^{K(\mu_{0},a,\delta ,\varepsilon_{\text{dist}})}\frac{(g_{\mu_{k + 1}}(W_{\mu_{k},\epsilon_{k}}) - g_{\mu_{k + 1}}(W_{\mu_{k + 1},\epsilon_{k + 1}}))}{\mu_{K}^{3}(\mu_{0},a,\delta ,\varepsilon_{\text{dist}})}\right) \\ = 2 (\mu_ {0} (a ^ {2} + 1) + 3 a ^ {2}) \left(\frac {1}{\beta^ {6} a ^ {6}} + \frac {1}{\mu_ {K (\mu_ {0} , a , \delta , \varepsilon_ {\mathrm {d i s t}})} ^ {3}}\right) \left(\sum_ {k = 0} ^ {K (\mu_ {0}, a, \delta , \varepsilon_ {\mathrm {d i s t}})} \left((g _ {\mu_ {k + 1}} (W _ {\mu_ {k}, \epsilon_ {k}}) - g _ {\mu_ {k + 1}} (W _ {\mu_ {k + 1}, \epsilon_ {k + 1}}))\right)\right) \\ \leq 2 (\mu_ {0} (a ^ {2} + 1) + 3 a ^ {2}) \left(\frac {1}{\beta^ {6} a ^ {6}} + \frac {1}{\mu_ {K (\mu_ {0} , a , \delta , \varepsilon_ {\mathrm {d i s t}})}}\right) \left(\sum_ {k = 0} ^ {K (\mu_ {0}, a, \delta , \varepsilon_ {\mathrm {d i s t}})} \left((g _ {\mu_ {k}} (W _ {\mu_ {k}} ^ {\epsilon_ {k}}) - g _ {\mu_ {k + 1}} (W _ {\mu_ {k + 1}} ^ {\epsilon_ {k + 1}}))\right)\right) \\ = 2 \left(\mu_ {0} \left(a ^ {2} + 1\right) + 3 a ^ {2}\right) \left(\frac {1}{\beta^ {6} a ^ {6}} + \frac {1}{\mu_ {K} ^ {3} \left(\mu_ {0} , a , \delta , \varepsilon_ {\text {d i s t}}\right)}\right) \left(\left(g _ {\mu_ {0}} \left(W _ {\mu_ {0}, \epsilon_ {0}}\right) - g _ {\mu_ {K} \left(\mu_ {0}, a, \delta , \varepsilon_ {\text {d i s t}}\right) + 1} \left(W _ {\mu_ {K} \left(\mu_ {0}, a, \delta , \varepsilon_ {\text {d i s t}}\right) + 1} ^ {\epsilon_ {K} \left(\mu_ {0}, a, \delta , \varepsilon_ {\text {d i s t}}\right) + 1}\right)\right) \right. \\ \leq 2 (\mu_ {0} (a ^ {2} + 1) + 3 a ^ {2}) \left(\frac {1}{\beta^ {6} a ^ {6}} + \frac {1}{\mu_ {K (\mu_ {0} , a , \delta , \varepsilon_ {\mathrm {d i s t}})}}\right) g _ {\mu_ {0}} (W _ {\mu_ {0}, \epsilon_ {0}}) \\ \end{array}
$$