SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \stackrel {(\Delta_ {3})} {\leq} \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {t}\right) \cdot \left(\frac {p _ {H} ^ {\star \star} - p _ {H} ^ {t}}{b _ {H} + c _ {H}}\right) + \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {t}\right) \cdot \left(\frac {p _ {L} ^ {\star \star} - p _ {L} ^ {t}}{b _ {L} + c _ {L}}\right) \\ - \eta^ {t} \Big (\underbrace {\operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {t}\right) \cdot G _ {H} (\mathbf {p} ^ {t} , \mathbf {p} ^ {t}) + \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {t}\right) \cdot G _ {L} (\mathbf {p} ^ {t} , \mathbf {p} ^ {t})} _ {\mathcal {G} (\mathbf {p} ^ {t})} \Big) \\ + 2 \eta^ {t} \cdot \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2} \\ \end{array}
$$
$$
\begin{array}{l} \stackrel {(\Delta_ {4})} {=} \frac {\left| p _ {H} ^ {\star \star} - p _ {H} ^ {t} \right|}{b _ {H} + c _ {H}} + \frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {t} \right|}{b _ {L} + c _ {L}} - \eta^ {t} \mathcal {G} (\mathbf {p} ^ {t}) + 2 \eta^ {t} \cdot \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2} \\ = \varepsilon (\mathbf {p} ^ {t}) - \eta^ {t} \left(\mathcal {G} (\mathbf {p} ^ {t}) - 2 \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2}\right) \\ \end{array}
$$
$$
\stackrel {(\Delta_ {5})} {\leq} \varepsilon (\mathbf {p} ^ {t}) - \eta^ {t} \left(M _ {\epsilon_ {0}} - 2 \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2}\right), \quad \forall t \geq T _ {1} ^ {\epsilon_ {0}}, \tag {29}
$$
where we use the property of the projection operator in  $(\Delta_1)$ . The equality  $(\Delta_2)$  follows since  $\mathrm{sign}\left(p_i^{\star \star} - p_i^{t + 1}\right) = \mathrm{sign}\left(p_i^{\star \star} - p_i^t\right)$  for  $i\in \{H,L\}$ , and the projection does not change the sign, then we have
$$
\mathrm {s i g n} \left(\frac {p _ {i} ^ {\star \star}}{b _ {i} + c _ {i}} - \eta^ {t} G _ {i} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t}) - \frac {p _ {i} ^ {t}}{b _ {i} + c _ {i}}\right) = \mathrm {s i g n} \left(p _ {i} ^ {\star \star} - p _ {i} ^ {t}\right), \quad \forall i \in \{H, L \}.
$$
The inequality  $(\Delta_3)$  results from Lemma F.5. In particular, since  $\left\| \nabla_{\mathbf{r}}G_i(\mathbf{p},\mathbf{r})\right\| _2\leq \ell_r$  , where  $\ell_r = (1 / 4)\sqrt{c_H^2 + c_L^2}$  , it follows that
$$
\left| G _ {i} \left(\mathbf {p} ^ {t}, \mathbf {r} ^ {t}\right) - G _ {i} \left(\mathbf {p} ^ {t}, \mathbf {p} ^ {t}\right) \right| \leq \left\| \nabla_ {\mathbf {r}} G _ {i} (\mathbf {p}, \mathbf {r}) \right\| _ {2} \cdot \left\| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \right\| _ {2} \leq \ell_ {r} \left\| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \right\| _ {2}, \quad \forall i \in \{H, L \}. \tag {30}
$$
In step  $(\Delta_4)$ , we introduce the function  $\mathcal{G}(\mathbf{p})$ , which is defined as
$$
\mathcal {G} (\mathbf {p}) := \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H}\right) \cdot G _ {H} (\mathbf {p}, \mathbf {p}) + \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L}\right) \cdot G _ {L} (\mathbf {p}, \mathbf {p}), \quad \forall \mathbf {p} \in \mathcal {P} ^ {2}. \tag {31}
$$
Finally, the last step  $(\Delta_5)$  is due to Lemma F.2, which states that  $\exists M_{\epsilon_0} > 0$  such that  $\mathcal{G}(\mathbf{p}) \geq M_{\epsilon_0}$  for every  $\mathbf{p} \in \mathcal{P}^2$  with  $\varepsilon(\mathbf{p}) > \epsilon_0$ . Since for  $t \geq T_1^{\epsilon_0} > T^{\epsilon_0}$ , we have  $\varepsilon(\mathbf{p}^t) \geq \epsilon_0$  by the premise, it follows that  $\mathcal{G}(\mathbf{p}^t) \geq M_{\epsilon_0}$ .
By Lemma F.1, the difference  $\{\mathbf{r}^t -\mathbf{p}^t\}_{t\geq 0}$  converges to 0 as  $t$  approaches to infinity. As a result, we can always find  $T_2^{\epsilon_0} > 0$  to ensure  $\| \mathbf{r}^t -\mathbf{p}^t\| _2\leq M_{\epsilon_0} / (4\ell_r)$  for all  $t\geq T_2^{\epsilon_0}$ . Then, the last line in Eq. (29) can be further upper-bounded as
$$
\begin{array}{l} \varepsilon (\mathbf {p} ^ {t + 1}) \leq \varepsilon (\mathbf {p} ^ {t}) - \eta^ {t} \left(M _ {\epsilon_ {0}} - 2 \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2}\right) \\ \leq \varepsilon (\mathbf {p} ^ {t}) - \eta^ {t} \left(M _ {\epsilon_ {0}} - \frac {1}{2} M _ {\epsilon_ {0}}\right) \tag {32} \\ \leq \varepsilon (\mathbf {p} ^ {t}) - \frac {1}{2} \eta^ {t} M _ {\epsilon_ {0}}, \quad \forall t \geq \widehat {T} ^ {\epsilon_ {0}} := \max  \left\{T _ {1} ^ {\epsilon_ {0}}, T _ {2} ^ {\epsilon_ {0}} \right\}. \\ \end{array}
$$
By a telescoping sum from any period  $T - 1 \geq \widehat{T}^{\epsilon_0}$  down to period  $\widehat{T}^{\epsilon_0}$ , we have that
$$
\varepsilon \left(\mathbf {p} ^ {T}\right) \leq \varepsilon \left(\mathbf {p} ^ {\widehat {T} ^ {\epsilon_ {0}}}\right) - \frac {M _ {\epsilon_ {0}}}{2} \sum_ {t = \widehat {T} ^ {\epsilon_ {0}}} ^ {T - 1} \eta^ {t}. \tag {33}
$$
Since the step-sizes satisfy  $\sum_{t=0}^{\infty} \eta^{t} = \infty$ , we have that  $\lim_{T \to \infty} (M_{\epsilon_{0}} / 2) \sum_{t=\widehat{T}^{\epsilon_{0}}}^{T-1} \eta^{t} = \infty$ , which further implies the contradiction that  $\lim_{T \to \infty} \varepsilon(\mathbf{p}^{T}) \to -\infty$ . Thus, the prices in consecutive periods will not always stay in the same region, i.e., the price path  $\{\mathbf{p}^{t}\}_{t \geq 0}$  keeps oscillating between different regions  $N_{j}$  for  $j \in \{1,2,3,4\}$ .
Next, we prove that  $\exists T_3^{\epsilon_0} \geq T^{\epsilon_0}$  such that the price path  $\{\mathbf{p}^t\}_{t \geq T_3^{\epsilon_0}}$  only oscillates between adjacent quadrants. Arguing by contradiction, we assume  $\eta^{T_3^{\epsilon_0}} < \epsilon_0 / (2M_G)$ , where  $M_G$  is the upper bound of  $|G_i(\mathbf{p}, \mathbf{r})|$  defined in Eq. (100) in Lemma F.5. If  $\mathbf{p}^t \in N_1$  (or  $N_2$ ) and  $\mathbf{p}^{t+1} \in N_3$  (or  $N_4$ ), i.e.,  $\text{sign}(p_i^{\star\star} - p_i^t) \neq \text{sign}(p_i^{\star\star} - p_i^{t+1})$  for both  $i \in \{H, L\}$ , then it automatically follows from the update rule (see Eq. (26)) that
$$
\begin{array}{l} \varepsilon (\mathbf {p} ^ {t + 1}) \leq \eta^ {t} \cdot \left(\left| G _ {H} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t}) \right| + \left| G _ {L} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t}) \right|\right) \\ \leq \frac {\epsilon_ {0}}{2 M _ {G}} \cdot 2 M _ {G} = \epsilon_ {0}, \tag {34} \\ \end{array}
$$
which again contradicts the premise that  $\{\mathbf{p}^t\}_{t\geq T^{\epsilon_0}}$  never visits  $N_{\epsilon_0}^1$ . Thus, when  $t\geq T_3^{\epsilon_0}$ , the price path  $\{\mathbf{p}^t\}_{t\geq T_3^{\epsilon_0}}$  only oscillates between adjacent quadrants.
Because of the following two established facts: (i) when  $t \geq \max \{T_2^{\epsilon_0}, T_3^{\epsilon_0}\}$ , the price path will not remain in the same quadrant but can only oscillate between adjacent quadrants, and (ii) the step-sizes  $\{\eta^t\}_{t \geq 0}$  decrease to 0 as  $t \to \infty$ , we conclude that the price path  $\{\mathbf{p}\}_{t \geq 0}$  must visit the set
$$
\widehat {N} _ {\epsilon_ {1}} := \left\{\mathbf {p} \in \mathcal {P} ^ {2} \left| \frac {\left| p _ {H} ^ {\star \star} - p _ {H} \right|}{b _ {H} + c _ {H}} <   \epsilon_ {1} \text {o r} \frac {\left| p _ {L} ^ {\star \star} - p _ {L} \right|}{b _ {L} + c _ {L}} <   \epsilon_ {1} \right. \right\} \tag {35}
$$
infinitely many times for any  $\epsilon_1 > 0$ . Intuitively, set  $\widehat{N}_{\epsilon_1}$  can be understood as the boundary regions between adjacent quadrants. We define  $\mathcal{T}_{\epsilon_1}$  as the set of "jumping periods" from  $(N_{1} \cup N_{3}) \cap \widehat{\widetilde{N}}_{\epsilon_{1}}$  to  $(N_{2} \cup N_{4}) \cap \widehat{N}_{\epsilon_{1}}$  after  $T^{\epsilon_0}$ , i.e.,
$$
\mathcal {T} _ {\epsilon_ {1}} := \left\{t \geq T ^ {\epsilon_ {0}} \mid \mathbf {p} ^ {t} \in \left(N _ {1} \cup N _ {3}\right) \cap \widehat {N} _ {\epsilon_ {1}} \text {a n d} \mathbf {p} ^ {t + 1} \in \left(N _ {2} \cup N _ {4}\right) \cap \widehat {N} _ {\epsilon_ {1}} \right\}. \tag {36}
$$
Then, the above fact  $(i)$  implies that  $|\mathcal{T}_{\epsilon_1}| = \infty$  for any  $\epsilon_{1} > 0$
The key idea of the following proof is that if  $\{\mathbf{p}^t\}_{t\geq 0}$  visits  $\widehat{N}_{\epsilon_1}$  with some sufficiently small  $\epsilon_{1}$  and a relatively smaller step-size, it will stay in  $\widehat{N}_{\epsilon_1}$  and converge to  $N_{\epsilon_0}^1$  concurrently. This results in a contradiction with our initial assumption that  $\{\mathbf{p}^t\}_{t\geq 0}$  never visits  $N_{\epsilon_0}^1$  when  $t\geq T^{\epsilon_0}$ .
Let  $T_1^{\epsilon_1} \in \mathcal{T}_{\epsilon_1}$  be a jumping period with  $\epsilon_{1} \ll \epsilon_{0}$ . Without loss of generality, suppose that  $|p_H^{\star \star} - p_H^{T_1^{\epsilon_1}}| / (b_H + c_H) < \epsilon_1$ . Then, since  $\mathbf{p}^{T_1^{\epsilon_1}} \notin N_{\epsilon_0}^1$ , we accordingly have that  $|p_L^{\star \star} - p_L^{T_1^{\epsilon_1}}| / (b_L + c_L) >$
$\epsilon_0 - \epsilon_1$  . By the update rule of price in Eq. (26), it holds that
$$
\begin{array}{l} \frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}} + 1} \right|}{b _ {L} + c _ {L}} = \mathrm {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}} + 1}\right) \cdot \frac {p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}} + 1}}{b _ {L} + c _ {L}} \\ = \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}} + 1}\right) \cdot \left(\frac {p _ {L} ^ {\star \star}}{b _ {L} + c _ {L}} - \operatorname {P r o j} _ {\mathcal {P} _ {L}} \left(\frac {p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}}{b _ {L} + c _ {L}} + \eta^ {T _ {1} ^ {\epsilon_ {1}}} G _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right)\right) \\ \leq \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}} + 1}\right) \cdot \left(\frac {p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}}{b _ {L} + c _ {L}} - \eta^ {T _ {1} ^ {\epsilon_ {1}}} G _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) \\ = \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot \frac {p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}}{b _ {L} + c _ {L}} - \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot \left(\eta^ {T _ {1} ^ {\epsilon_ {1}}} G _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right). \tag {37} \\ \end{array}
$$
We note that, the last step above is due to the premise that the step-size is sufficiently small: since  $\left| p_{L}^{\star \star} - p_{L}^{T_{1}^{\epsilon_{1}}} \right| / (b_{L} + c_{L}) > \epsilon_{0} - \epsilon_{1}$ , the equality sign  $\left( p_{L}^{\star \star} - p_{L}^{T_{1}^{\epsilon_{1}} + 1} \right) = \text{sign} \left( p_{L}^{\star \star} - p_{L}^{T_{1}^{\epsilon_{1}}} \right)$  holds if  $\eta^{T_{1}^{\epsilon_{1}}} < (\epsilon_{0} - \epsilon_{1}) / M_{G}$ . Next, we show that the second term on the right-hand side of Eq. (37) is upper-bounded away from zero:
$$
\operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot G _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)
$$
$$
\geq \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot G _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) - \left| G _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) - G _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \right|
$$
$$
\begin{array}{l} \stackrel {(\Delta_ {1})} {\geq} \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot G _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) - \ell_ {r} \left\| \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}} \right\| _ {2} \\ = \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot \left[ \frac {1}{\left(b _ {L} + c _ {L}\right) p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}} + d _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) - 1 \right] - \ell_ {r} \left\| \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}} \right\| _ {2} \\ \geq \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot \left[ \frac {1}{\left(b _ {L} + c _ {L}\right) p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}} + d _ {L} \left(\left(p _ {H} ^ {\star \star}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right), \left(p _ {H} ^ {\star \star}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) - 1 \right] \\ - \left| d _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) - d _ {L} \left(\left(p _ {H} ^ {\star \star}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right), \left(p _ {H} ^ {\star \star}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) \right| - \ell_ {r} \left\| \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}} \right\| _ {2} \\ \end{array}
$$
$$
\begin{array}{l} \stackrel {(\Delta_ {2})} {\geq} \operatorname {s i g n} \cdot \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot \left[ \frac {1}{\left(b _ {L} + c _ {L}\right) p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}} + d _ {L} \left(\left(p _ {H} ^ {\star \star}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right), \left(p _ {H} ^ {\star \star}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) - 1 \right] - \ell_ {r} \left\| \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}} \right\| _ {2} \\ - \max  _ {\mathbf {p} \in \mathcal {P}} \left\{\left| \frac {\partial d _ {L} (\mathbf {p} , \mathbf {p})}{\partial p _ {H}} \right| \right\} \cdot \left| p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}} \right| \\ \end{array}
$$
$$
\begin{array}{l} \stackrel {(\Delta_ {3})} {\geq} \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot \left[ \frac {1}{(b _ {L} + c _ {L}) p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}} + d _ {L} \left((p _ {H} ^ {\star \star}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}), (p _ {H} ^ {\star \star}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}})\right) - 1 \right] - \ell_ {r} \left\| \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}} \right\| _ {2} \\ - \frac {1}{4} b _ {H} \left(b _ {H} + c _ {H}\right) \epsilon_ {1} \\ = \mathcal {G} \left(\left(p _ {H} ^ {\star \star}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) - \ell_ {r} \left\| \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}} \right\| _ {2} - \frac {1}{4} b _ {H} \left(b _ {H} + c _ {H}\right) \epsilon_ {1}, \tag {38} \\ \end{array}
$$
where  $(\Delta_1)$  holds because of the mean value theorem and the fact that  $\| \nabla_{\mathbf{r}}G_L(\mathbf{p},\mathbf{r})\| _2\leq \ell_r$  for  $\mathbf{p},\mathbf{r}\in \mathcal{P}^2$  by Lemma F.5, inequality  $(\Delta_{2})$  applies the mean value theorem again to the demand function. In  $(\Delta_3)$ , we use the assumption that  $\left|p_H^{\star \star} - p_H^{\frac{T^\epsilon 1}{1}}\right| < (b_H + c_H)\epsilon_1$  and apply a similar
argument as Eq. (101) to derive that
$$
\left| \frac {\partial d _ {L} (\mathbf {p} , \mathbf {p})}{\partial p _ {H}} \right| = \left| b _ {H} \cdot d _ {L} (\mathbf {p}, \mathbf {p}) \cdot d _ {H} (\mathbf {p}, \mathbf {p}) \right| \leq \frac {1}{4} b _ {H}, \quad \forall \mathbf {p} \in \mathcal {P} ^ {2}. \tag {39}
$$
Since  $\varepsilon\big((p_H^{\star \star}, p_L^{T_1^{\epsilon_1}})\big) > \epsilon_0 - \epsilon_1$ , by Lemma F.2, there exists a constant  $M_{\epsilon_0 - \epsilon_1} > 0$ , such that  $\mathcal{G}\big((p_H^{\star \star}, p_L^{T_1^{\epsilon_1}})\big) \geq M_{\epsilon_0 - \epsilon_1}$ . Meanwhile, we can choose  $\epsilon_1$  sufficiently smaller than  $\epsilon_0$  to ensure that  $(1/2)M_{\epsilon_0 - \epsilon_1} - (1/4)b_H(b_H + c_H)\epsilon_1 > 0$ . Furthermore, recall that  $T_1^{\epsilon_1}$  can be arbitrarily chosen from  $\mathcal{T}_{\epsilon_1}$ , where  $|\mathcal{T}_{\epsilon_1}| = \infty$ . Thus, we can always find a sufficiently large  $T_1^{\epsilon_1} \in \mathcal{T}_{\epsilon_1}$  to guarantee that  $\ell_r\| \mathbf{r}^{T_1^{\epsilon_1}} - \mathbf{p}^{T_1^{\epsilon_1}}\|_2 \leq (1/2)M_{\epsilon_0 - \epsilon_1} - (1/4)b_H(b_H + c_H)\epsilon_1$  by Lemma F.1. Back to Eq. (38), it follows that
$$
\begin{array}{l} \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot G _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \\ \geq \mathcal {G} \left(\left(p _ {H} ^ {\star \star}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) - \ell_ {r} \left\| \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}} \right\| _ {2} - \frac {1}{4} b _ {H} \left(b _ {H} + c _ {H}\right) \epsilon_ {1} \\ \geq M _ {\epsilon_ {0} - \epsilon_ {1}} - \left(\frac {1}{2} M _ {\epsilon_ {0} - \epsilon_ {1}} - \frac {1}{4} b _ {H} \left(b _ {H} + c _ {H}\right) \epsilon_ {1}\right) - \frac {1}{4} b _ {H} \left(b _ {H} + c _ {H}\right) \epsilon_ {1} \tag {40} \\ = \frac {1}{2} M _ {\epsilon_ {0} - \epsilon_ {1}}. \\ \end{array}
$$
By substituting Eq. (40) into Eq. (37), we further derive that
$$
\begin{array}{l} \frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}} + 1} \right|}{b _ {L} + c _ {L}} \leq \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot \frac {p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}}{b _ {L} + c _ {L}} - \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot \left(\eta^ {T _ {1} ^ {\epsilon_ {1}}} G _ {L} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) \\ \leq \frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}} \right|}{b _ {L} + c _ {L}} - \frac {1}{2} \eta^ {T _ {1} ^ {\epsilon_ {1}}} M _ {\epsilon_ {0} - \epsilon_ {1}}. \tag {41} \\ \end{array}
$$
This indicates that the price update from  $p_{L}^{T_{1}^{\epsilon_{1}}}$  to  $p_{L}^{T_{1}^{\epsilon_{1}} + 1}$  is towards the SNE price  $p_{L}^{\star \star}$  when  $|p_{H}^{\star \star} - p_{H}^{T_{1}^{\epsilon_{1}}}| < (b_{H} + c_{H})\epsilon_{1}$ .
Now, we use a strong induction to show that  $\left|p_H^{\star \star} - p_H^t\right| / (b_H + c_H) < \epsilon_1$  holds for all  $t \geq T_1^{\epsilon_1}$  provided that  $\epsilon_1$  is sufficiently small and  $T_1^{\epsilon_1}$  is sufficiently large. Suppose there exists  $T_2^{\epsilon_1} \geq T_1^{\epsilon_1}$  such that  $\left|p_H^{\star \star} - p_H^t\right| / (b_H + c_H) < \epsilon_1$  holds for all  $t = T_1^{\epsilon_1}, T_1^{\epsilon_1} + 1, \ldots, T_2^{\epsilon_1}$ , we aim to show that  $\left|p_H^{\star \star} - p_H^{T_2^{\epsilon_1} + 1}\right| / (b_H + c_H) < \epsilon_1$ . Following the same derivation from Eq. (37) to Eq. (41), it holds that
$$
\frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {t + 1} \right|}{b _ {L} + c _ {L}} \leq \frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {t} \right|}{b _ {L} + c _ {L}} - \frac {1}{2} \eta^ {t} M _ {\epsilon_ {0} - \epsilon_ {1}}, \quad t = T _ {1} ^ {\epsilon_ {1}}, T _ {1} ^ {\epsilon_ {1}} + 1, \dots , T _ {2} ^ {\epsilon_ {1}}. \tag {42}
$$
By telescoping the above inequality from  $T_2^{\epsilon_1} - 1$  down to  $T_1^{\epsilon_1}$ , we have that
$$
\frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {2} ^ {\epsilon_ {1}}} \right|}{b _ {L} + c _ {L}} \leq \frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}} \right|}{b _ {L} + c _ {L}} - \frac {M _ {\epsilon_ {0} - \epsilon_ {1}}}{2} \sum_ {t = T _ {1} ^ {\epsilon_ {1}}} ^ {T _ {2} ^ {\epsilon_ {1}} - 1} \eta^ {t} \leq \frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}} \right|}{b _ {L} + c _ {L}}. \tag {43}
$$
Additionally, we note that  $\left|p_H^{\star \star} - p_H^t\right| / (b_H + c_H) < \epsilon_1$  and  $\mathbf{p}^t \notin N_{\epsilon_0}^1$  together imply that  $\left|p_L^{\star \star} - p_L^t\right| / (b_L + c_L) > (\epsilon_0 - \epsilon_1)$  for all  $t = T_1^{\epsilon_1}, T_1^{\epsilon_1} + 1, \ldots, T_2^{\epsilon_1}$ . As the step-size is sufficiently small, we conclude that the price path  $\{p_L^t\}_{t = T_1^{\epsilon_1}}^{T_2^{\epsilon_1}}$  lies on the same side of the SNE price  $p_L^{\star \star}$ , and thus sign  $\left(p_L^{\star \star} - p_L^{T_2^{\epsilon_1}}\right) = \mathrm{sign}\left(p_L^{\star \star} - p_L^{T_1^{\epsilon_1}}\right)$ . Below, we discuss two cases based on the value of  $\left|p_H^{\star \star} - p_H^{T_2^{\epsilon_1}}\right|$ :
Case 1:  $\left| p_H^{\star \star} - p_H^{T_2^{\epsilon 1}} \right| / (b_H + c_H) < \epsilon_1 - \eta^{T_1^{\epsilon 1}} M_G.$
Following a similar derivation as Eq.(37), we have that
$$
\begin{array}{l} \frac {\left| p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}} + 1} \right|}{b _ {H} + c _ {H}} \leq \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot \frac {p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}}{b _ {H} + c _ {H}} - \eta^ {T _ {2} ^ {\epsilon_ {1}}} \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}}\right). \\ \leq \frac {\left| p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}} \right|}{b _ {H} + c _ {H}} + \eta^ {T _ {2} ^ {\epsilon_ {1}}} \left| G _ {H} \left(\mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \right| \\ \stackrel {(\Delta_ {1})} {<  } \left(\epsilon_ {1} - \eta^ {T _ {1} ^ {\epsilon_ {1}}} M _ {G}\right) + \eta^ {T _ {2} ^ {\epsilon_ {1}}} M _ {G} \\ \begin{array}{l} \left(\Delta_ {2}\right) \\ \leq \epsilon_ {1}, \end{array} \tag {44} \\ \end{array}
$$
where  $(\Delta_1)$  is from Lemma F.5 that  $|G_H(\mathbf{p},\mathbf{r})|\leq M_G$  for  $\mathbf{p},\mathbf{r}\in \mathcal{P}^2$ , and  $(\Delta_{2})$  is because the sequence of step-sizes is non-increasing, hence  $\eta^{T_1^{\epsilon_1}}\geq \eta^{T_2^{\epsilon_1}}$ .
Case 2:  $\left| p_H^{\star \star} - p_H^{T_2^{\epsilon_1}} \right| / (b_H + c_H) \in \left[ \epsilon_1 - \eta^{T_1^{\epsilon_1}} M_G, \epsilon_1 \right)$ .
It suffices to show that  $\left| p_{H}^{\star \star} - p_{H}^{T_{2}^{\epsilon 1} + 1} \right| \leq \left| p_{H}^{\star \star} - p_{H}^{T_{2}^{\epsilon 1}} \right|$ . According to the observation below Eq. (27), this is equivalent to showing that
$$
\operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \geq 0. \tag {45}
$$
We further discuss two scenarios depending on whether  $\mathbf{p}^{T_2^{\epsilon 1}}\in N_1\cup N_3$  or  $\mathbf{p}^{T_2^{\epsilon 1}}\in N_2\cup N_4$