Establishing almost everywhere convergence of the Bochner-Riesz operator is of fundamental interest in harmonic analysis. In 1961, Stein proved that the problem is strongly related to the problem of obtaining a weak Lp estimate of the Bochner-Riesz operator. Now the result is called Stein's maximal principle. In this talk, we introduce a brief history on the almost everywhere convergence problem and give a statement of Stein's maximal principle. We also study how he proved this result.