The Gromov-Hausdorff metric space $(\mathcal{M},d_{GH})$ is complete. I'm currently following the proof of this fact given in Petersen's Riemannian Geometry (3rd Edition) (see Proposition 11.1.8). Given a Cauchy sequence $\{X_n\}$ of compact metric spaces, it suffices to show that there's some convergent subsequence $\{X_i\}$ (I'll mainly follow the notations in Petersen). My question concerns the limit space of this subsequence. To state the question let me first describe briefly the limit space.
Define \begin{align} \hat{X}=\big\{\{x_i\}:x_i\in X_i\text{ and }\{x_i\}\text{ is Cauchy in }Y\big\} \end{align} where $Y:=\sqcup_{i}X_i$ is the disjoint union of all $X_i$'s with metric \begin{align} |x_ix_{i+j}|=\inf_{\{x_{i+k}\in X_{i+k}\}}\sum_{k=1}^j|x_{i+k-1}x_{i+k}| \end{align} (The subsequence $\{X_i\}$ can be chosen to satisfies $d_{GH}(X_i,X_{i+1})<2^{-i}$ for all $i$. From this one can choose an admissible metric (following Petersen's terminology) on $X_i\sqcup X_{i+1}$ such that $d_H(X_i,X_{i+1})\leq 2^{-i}$. The metric on $Y$ is defined using these admissible metrics.) Define \begin{align} |\{x_i\}\{y_i\}|=\lim_{i\to\infty}|x_iy_i| \end{align} This gives a pseudo-metric on $\hat{X}$. By taking quotient we obtain a metric space $X$.
In the proof there, it is proved that $\{X_i\}$ converges to $X$. However, it is not immediately clear to me why in the first place that the limit space $X$ is compact, and the proof there does not say anything about it. Thus I decide to prove it by myself. I have proved that $X$ is complete. My question is
How do we show that $X$ is totally bounded?
Since each $X_i$ is compact, thus totally bounded, given any $\epsilon>0$ we can find an $N_i=N_i(\epsilon)\in\mathbb{N}$ (depending on both $i$ and $\epsilon$) and a subset $S_i(\epsilon)=\{x_{i,\alpha}\}_{\alpha=1}^{N_i}$ of $X_i$ such that \begin{align} X_i=\bigcup_{\alpha=1}^{N_i}B(x_{i,\alpha},\epsilon) \end{align} I've observed that due to the condition $d_H(X_i,X_{i+1})\leq 2^{-i}$, the $N_i$ can actually be chosen to be independent of $i$ and depends only on $\epsilon$. Henceforth let me write it simply as $N$ and let us assume all $S_i(\epsilon)$ have $N$ elements. My attempt now is to show that for each $1\leq\alpha\leq N$, the sequence $\{x_{i,\alpha}\}_{i}$ is Cauchy in $Y$, and then show that the (equivalence classes of) sequences $\{x_{i,\alpha}\}_{i}$ form the desired finite subset for $X$ to be covered by $\epsilon$-balls centered at them. However, I have failed to even show that $\{x_{i,\alpha}\}_{i}$ is Cauchy.
Any comment or answer is greatly appreciated.