The purpose of this note is to indicate a direct and natural way of proving theorems stated in  by using an explicit expression for the sequence of normalizing constants belonging to a distribution function attracted to a stable law. This results in a remark concerning a counter example given by Tucker and a slightly sharpened version of his Lemma 5.
Laurens De Haan. "Note on a Paper by H. G. Tucker." Ann. Math. Statist. 41 (2) 729 - 732, April, 1970. https://doi.org/10.1214/aoms/1177697124