\hat{\Gamma}(j,j)} > \sqrt{\underline{\sigma}^2 - C \aleph_n} >0$ and, by Weyl’s inequality, the minimal eigenvalue of $\hat{V}$ is no smaller than $v - C
\daleth_n > 0$. In particular, the error terms $\Delta^*_{n,1}$ and $\Delta^*_{n,2}$ are well-defined (i.e. positive). Thus we have that $$\label{eq:A3}
A_3 \leq C \left( \Delta^*_{n,1} + \Delta^*_{n,2} \right) +
\frac{2}{n},$$ where the lower order term $\frac{1}{n}$ is reported to account for the restriction to the event $\mathcal{E}_n$. The result now follows by combining all the bounds, after noting that $\Delta_{1,n} \leq \Delta^*_{1,n}$ and $\Delta_{2,n} \leq \Delta^*_{2,n}$.

To show that the same bound holds for the coverage of $\tilde{C}^*_\alpha$ we proceed in a similar manner. Using the triangle inequality, and uniformly over all the distributions in ${\cal P}_n$, $$\begin{aligned}
\mathbb{P}(\theta \in \tilde{C}^*_n)  & =
\mathbb{P}(\sqrt{n} |\hat\theta_j - \theta _j| \leq  \tilde{t}^*_{j,\alpha},
\forall j)\\
& \geq
\mathbb{P}\left( \sqrt{n} |\hat \theta^*_j - \hat{\theta}_j | \leq
\tilde{t}^*_{j,\alpha}, \forall j \Big| (W_1,\ldots,W_n) \right) - (A_1 + A_2 +
A_3)\\
& \geq   (1 - \alpha) - (A_1 + A_2 +
A_3),\end{aligned}$$ where $$\begin{aligned}
    A_1 & = \sup_{t = (t_1,\ldots,t_s) \in \mathbb{R}_+^s} \left| \mathbb{P}\left( \sqrt{n} |
    \hat{\theta}_j-
    \theta_j   |
    \leq t_j, \forall j \right) - \mathbb{P}( |Z_{n,j}| \leq t_j, \forall j ) \right|,\\
    A_2 & = \sup_{t = (t_1,\ldots,t_s) \in \mathbb{R}_+^s } \left|  \mathbb{P}(
    | Z_{n,j} | \leq t_j, \forall j ) - \mathbb{P}( |
    \hat{Z}_{n,j} | \leq t_j, \forall j )      \right|,\\
    \text{and} & \\
    A_3 & = \sup_{t = (t_1,\ldots,t_s) \in \mathbb{R}_+^s   } \Big|  \mathbb{P}(
    | \hat{Z}_{n,j} | \leq
    t_j, \forall j) -  \mathbb{P}\left( \sqrt{n} | \hat{\theta}_j^* -
    \hat{\theta}_j   | \leq t_j, \forall j
   \Big| (W_1,\ldots,W_n) \right)  \Big|.\end{aligned}$$ The term $A_1$ is bounded by $C (\Delta_{1,n} + \Delta_{2,n})$, as shown in the first part of the proof 