Retrieve shortest path between two nodes using Bellman-Ford-Moore algorithm sequentially
The correctness proof of the Bellman–Ford algorithm relies on the following lemma (compare Wikipedia):


  After $i$ iterations, $d[v]$ is at most the distance of the shortest path of length $i$ from the source vertex to $v$.


Indeed, if we perform all updates at a specific iteration in parallel, then $d[v]$ equals the distance of the shortest path of length $i$. When we perform the updates sequentially we might get lucky, and this depends on the order of vertices in the path.

This shows that in general you cannot stop Bellman–Ford early in the way you envision.