The problem I am trying to solve is the following.
Let $P = (u=v_0,v_1, \cdots ,v_k = v),k\geq 1$,be a $u-v$ geodesic in a connected graph $G$. Prove that $d(u,v_i) = i$ for each integer $i$ with $1\leq i\leq k$.
My proof is as follows.
Notice that since $P$ is a geodesic, $d(u,v) = k.$ Now, $d(u,v_i)\leq i$ since there exists a walk with distance $i$ and thus a path with distance at most $i$. However, if $d(u,v_i)<i$, then this would imply that $P$ is not minimal, a contradiction. Thus, $d(u,v_i)=i$.
Could someone check my proof for me, or maybe provide me with hints on how to correct my proof? Or is there a better/faster proof out there that someone could share with me?