The question doesn’t clarify that the iterations are for selecting the correct number of centroids. If we have a fixed value of K and run the K-means algorithm for, say, 100 iterations, I’m confused about why the cost would never increase.

In addition, if the centroids are already in an optimal position where moving them would increase the distance to their assigned points, the algorithm will leave the centroids unchanged and thus the cost will remain the same. It may plateau, indicating that the algorithm has reached a local minimum or a stable configuration of centroids where further movement would not reduce the cost, where the algorithm stops if a convergence criterion is met (e.g., cost changes below a certain threshold).