Demonstrating the Convergence of k-Means Clustering

The k-Means algorithm is a popular method for partitioning a dataset into k distinct clusters, where k is predetermined by the user. Upon completion, each data point is assigned to exactly one cluster. Regardless of the initial choice of k, the algorithm is guaranteed to reach a point where no further changes occur, meaning that convergence is achieved, and each data point remains in its designated cluster. Construct a logical proof to demonstrate that the k-Means clustering algorithm will reach convergence within a finite number of iterations. This proof need not reflect the most efficient or practical real-world implementation. The focus is solely on proving finite convergence. Include any necessary assumptions for ensuring the algorithm's convergence.