Let $X_1,X_2,...$ be a sequence of compact spaces. Assume $X_n\rightarrow X$ with respect to the Gromov Hausdorff metric, where $X$ is compact. If $\gamma:I\rightarrow X$ is a curve in $X$, does it follow that there exists a sequence of curves $\gamma_n:I\rightarrow X_n$
such that $d(\gamma(1),\gamma(0))=\lim_n d_n(\gamma_n(1),\gamma_n(0))$ ?
It seems intuitively reasonable since each $d_{GH}(X_n,X)\rightarrow 0$ means that for sufficiently large $n$, $X_n$ and $X$ become more and more isometric. Though the construction of each $\gamma_n$ doesn't seem to be clear to me.
Edit:
One can argue that for $x_1,x_2\in X$, we can find, for large enough $n$, $x_1^n,x_2^n \in X_n$ such that $d_n(x_1^n,x_2^n)\rightarrow d(x_1,x_2)$. Thus, we can do this for any finite subset $F$ of $X$.
However, I am not sure if $\gamma_n$ exist.
I suppose if each $X_n$ is path connected then we may do this due to my remark above.
You more or less answered your own question. Let $x^n_0, x^n_1 \in X_n$ so that $$\tag{1} d_n (x^n_0, x^n_1) \to d(\gamma(0), \gamma(1)).$$ For each $n\in \mathbb N$, since $X_n$ is path connected, there is a continuous curve $\gamma_n : I \to \mathbb R$ so that $\gamma_n(0) = x^n_0$ and $\gamma_n(1) = x^n_1$. Hence (1) is the equation you want. Of course $\gamma_n$ is very arbitrary here.