$$
\begin{array}{l} Z _ {c} \triangleq \sum_ {t = 1} ^ {T} c _ {t} \left(\bar {V} _ {t} ^ {s _ {t + 1}} - \underline {{V}} _ {t} ^ {s _ {t + 1}}\right) \\ \leq \sum_ {t = 1} ^ {T} c _ {t} \left(\bar {V} _ {t + 1} ^ {s _ {t + 1}} - \underline {{V}} _ {t + 1} ^ {s _ {t + 1}}\right) + \mathcal {O} \left(c _ {\max } \sum_ {s} \sum_ {t = 1} ^ {T} \left(\left| \bar {V} _ {t + 1} ^ {s} - \bar {V} _ {t} ^ {s} \right| + \left| \underline {{V}} _ {t + 1} ^ {s} - \underline {{V}} _ {t} ^ {s} \right|\right)\right) \\ \leq \sum_ {t = 1} ^ {T} c _ {t - 1} \left(\bar {V} _ {t} ^ {s _ {t}} - \underline {{V}} _ {t} ^ {s _ {t}}\right) + \mathcal {O} \left(c _ {\max } \sum_ {s} \sum_ {i = 1} ^ {T} \frac {\alpha_ {i}}{1 - \gamma}\right) \quad (\text {s h i f t i n g t h e i n d i c e s a n d d e f i n e} c _ {0} = 0) \\ \leq \sum_ {t = 1} ^ {T} c _ {t - 1} \left(\tilde {V} _ {t} ^ {s _ {t}} - \tilde {V} _ {t} ^ {s _ {t}}\right) + \mathcal {O} \left(\frac {c _ {\max } S}{1 - \gamma} \times H \ln T\right) \quad (\text {u s i n g} \alpha_ {i} = \frac {H + 1}{H + i}) \\ = \sum_ {s} \sum_ {\tau = 1} ^ {n _ {T + 1} (s)} c _ {t _ {\tau} (s) - 1} \left(\tilde {V} _ {t _ {\tau} (s)} ^ {s} - V _ {t _ {\tau} (s)} ^ {s}\right) + \mathcal {O} \left(\frac {c _ {\max} S \ln^ {2} T}{(1 - \gamma) ^ {2}}\right) \qquad (H = \frac {\ln T}{1 - \gamma}) \\ = \gamma \sum_ {s} \sum_ {\tau = 1} ^ {n _ {T + 1} (s)} c _ {t _ {\tau} (s) - 1} \sum_ {i = 1} ^ {\tau - 1} \alpha_ {\tau - 1} ^ {i} \left(\bar {V} _ {t _ {i} (s)} ^ {s _ {t _ {i} (s) + 1}} - \underline {{V}} _ {t _ {i} (s)} ^ {s _ {t _ {i} (s) + 1}} + 2 \mathsf {b n s} _ {i}\right) + \mathcal {O} \left(\frac {c _ {\max } S \ln^ {2} T}{(1 - \gamma) ^ {2}}\right) \\ \leq \gamma \sum_ {s} \sum_ {i = 1} ^ {n _ {T + 1} (s) - 1} \underbrace {\left(\sum_ {\tau = i + 1} ^ {n _ {T + 1} (s)} \alpha_ {\tau - 1} ^ {i} c _ {t _ {\tau} (s) - 1}\right)} _ {c _ {t _ {i} (s)} ^ {\prime}} \left(\overline {{V}} _ {t _ {i} (s)} ^ {s _ {t _ {i} (s) + 1}} - \underline {{V}} _ {t _ {i} (s)} ^ {s _ {t _ {i} (s) + 1}}\right) \\ + \mathcal {O} \left(\sum_ {s} \sum_ {\tau = 2} ^ {n _ {T + 1} (s)} c _ {t _ {\tau} (s) - 1} \mathsf {b n s} _ {\tau - 1} + \frac {c _ {\max } S \ln^ {2} T}{(1 - \gamma) ^ {2}}\right) \\ \leq \gamma \sum_ {t = 1} ^ {T} c _ {t} ^ {\prime} \left(\bar {V} _ {t} ^ {s _ {t + 1}} - \underline {{V}} _ {t} ^ {s _ {t + 1}}\right) + \mathcal {O} \left(\sum_ {s} \sum_ {\tau = 1} ^ {C _ {s} / c _ {\max }} c _ {\max } \mathsf {b n s} _ {\tau} + \frac {c _ {\max } S \ln^ {2} T}{(1 - \gamma) ^ {2}}\right) (C _ {s} \triangleq \sum_ {\tau = 1} ^ {n _ {T + 1} (s)} c _ {t _ {\tau} (s) - 1}) \\ \leq \gamma \sum_ {t = 1} ^ {T} c _ {t} ^ {\prime} \left(\bar {V} _ {t} ^ {s _ {t + 1}} - \underline {{V}} _ {t} ^ {s _ {t + 1}}\right) \\ + \mathcal {O} \left(\sum_ {s} \sum_ {\tau = 1} ^ {C _ {s} / c _ {\max}} \frac {c _ {\max} A \ln^ {2} (A S T / \delta) (\beta + \alpha_ {\tau} / \eta)}{(1 - \gamma) ^ {2}} + \frac {c _ {\max} \ln^ {2} T}{(1 - \gamma) ^ {2}}\right) \\ \leq \gamma \sum_ {t = 1} ^ {T} c _ {t} ^ {\prime} \left(\bar {V} _ {t} ^ {s _ {t + 1}} - \underline {{V}} _ {t} ^ {s _ {t + 1}}\right) + \mathcal {O} \left(\frac {C A \ln^ {2} (A S T / \delta) \beta}{(1 - \gamma) ^ {2}} + \frac {c _ {\max } A S \ln^ {3} (A S T / \delta)}{\eta (1 - \gamma) ^ {2}}\right). \tag {19} \\ \end{array}
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