Prompt: "no" Result: no}\ \ \text{otherwise}\\ \end{array} \right.$$ Since $\max_{i \in \left[m\right]} \left\| {\mathbf{x}}_i - {\mathbf{x}}_0 \righ

Prompt: "no" Result: no}\ \ \text{otherwise}\\ \end{array} \right.$$ Since $\max_{i \in \left[m\right]} \left\| {\mathbf{x}}_i - {\mathbf{x}}_0 \right\|_2 = \epsilon$, we need to upper bound the probabilities $\mathbb{P} \left( \min_{i \in \left[m\right]} \left\| {\mathbf{x}}_i - {\mathbf{x}}_0 \right\|_2 < \epsilon \right)$. We denote this probability by $P_1(\epsilon)$, since it is the first term of $\mathbb{P} \left( \max_{i \in \left[m\right]} \left\| {\mathbf{x}}_i - {\mathbf{x}}_0 \right\|_2 < \epsilon \right)$. Given a covering of size $B$, we need to bound the probability $\mathbb{P} \left( \min_{i \in \left[m\right]}