Reduced k-means clustering is a method for clustering objects in a low-dimensional subspace. The advantage of this method is that both clustering of objects and low-dimensional subspace reflecting the cluster structure are simultaneously obtained. In this paper, the relationship between conventional k-means clustering and reduced k-means clustering is discussed. Conditions ensuring almost sure convergence of the estimator of reduced k-means clustering as unboundedly increasing sample size have been presented. The results for a more general model considering conventional k-means clustering and reduced k-means clustering are provided in this paper. Moreover, a consistent selection of the numbers of clusters and dimensions is described.