Detecting Negative Cycles

After $n-1$ iterations, all shortest paths are finalized if no negative cycle exists. Any simple path has been fully explored. Run one more iteration (the $n$-th round). If any distance improves, a negative cycle exists. Why?

Because improving a distance in the $n$-th round means there is a path with $n$ or more edges that is better than the path with $n-1$ edges.

A path with $n$ edges in a graph with $n$ nodes must contain a cycle. If this path is shorter, the cycle must have negative total weight. You can traverse this cycle repeatedly to make the path arbitrarily short.