We show that the idea used by Kempf (1990) in order to obtain a splitting criterion for vector bundles on projective spaces leads to an elementary proof of the Babylonian tower theorem for this class of bundles, a result due to Barth and Van de Ven (1974) in the rank 2 case and to Sato (1977,1978) and Tyurin (1976) in the case of arbitrary rank. As a byproduct, we obtain a slight improvement of the numerical criterion of Flenner (1985) in the particular case under consideration.
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