If the longest loop free path in the network is N, then the Bellman Ford routing algorithm will converge to the lowest cost spanning tree after N iterations.

This is true, but

What happen if they converge before N iterations?

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Best answer

If the algorithm converges early, then it will simply continue to run (each node sharing its distance vector values with its neighbors), but none of the distance values will change.

The point is that after N rounds, everyone WILL know the shortest distances to each other nodes (assuming that no links/nodes are changing).