A convergent sequence of geodesics converges to a geodesic I'm reading Riemannian Geometry and Geometric Analysis by Jurgen Jost, and having a question about part of the following theorem.

Theorem 1.5.1 Let $M$ be a compact Riemannian manifold, $p,q \in M$. Then there exists a geodesic in every homotopy class of curves from $p$ to $q$, and this geodesic may be chosen as a shortest curve in its homotopy class.

Before outlining the proof of the theorem, we need a Corollary made before.

Corollary 1.4.4. Let $M$ be a compact Riemannian manifold. Then there exists $\rho_0>0$ with the property that any two points $p,q\in M$ with $d(p,q)\le \rho_0$ can be connected by precisely one geodesic of shortest length. This geodesic depends continuously on $p$ and $q$.

Outline of Theorem 1.5.1
Take a minimizing sequence $(\gamma_n)_{n \in \mathbb{N}}$ for arc length in the given homotopy class. We may assume that for any $\gamma_n$, there exist points $p_{0,n},...,p_{m,n}$ for which $d(p_{j-1,n},p_{j,n}) \le \rho_0$, where $m$ is independent of $\gamma_n$ and $\rho_0 <0$ is the constant in Corollary 1.4.4.
By compactness, after selection of a subsequence (still denote $\gamma_n, p_{j,n}$), the point $p_{j,n}$ converges to points $p_j$, and the segment of $\gamma_n$ between $p_{j-1,n},p_{j,n}$ converges to the shortest geodesic between $p_{j-1},p_{j}$.
Question
In the book, it says the statement in bold can be seen by using Corollary 1.4.4. but I don't quite understand what does it mean. Does the continuous dependency on endpoints directly imply the statement in bold? My attempt is to, on each segment, view the geodesics as functions $\gamma_n: [0,1] \to M$ for all $n$ and justify the limit function $\gamma: [0,1] \to M$ also satisfies the geodesic equation. But I have some trouble justifying this.
 A: Looking at the differential equation would, presumable, repeat the hard work you will have put into proving the Corollary 1.4.4. I'm not sure what you have available at that point, but probably everything you need to follow this reasoning (to simplify notation fix one index $j$ and set $p := p_j, p_n = p_{j,n}$ etc):
Have a look at the exponential maps in $p$ and $p_n$ and the geodesic segments $\gamma_n$ under $\exp_{p_n}$. They will be of the form $t\mapsto exp_{p_n}t v_n$ for some sequence of unit vectors $v_n \in T_{p_n}M$.
After possibly choosing another subsequence the initial vectors $v_n$  will converge to some vector $v$ of length one in $T_pM$ and the curve segments to the segment $exp_ptv$. This is using the continuity of $exp$ on $TM$ in a neighbourhood of the zero section -- and the compactness of the unit sphere (bundle). At this point you may want to have another closer look and some additional reasoning if this is not clear to you. Now you need to know that $exp_ptv $ is, for small $t$, the locally unique geodesic from $p$ to $exp_ptv$ ('small $t$' to be chosen in such a way that you are covered by the Corollary 1.4.4), and you are done.
So continuity of the geodesics should be understood in the sense of continuity of $\exp_pv$.
