TITLE: Tightness of a sequence of probability measures and weak convergence of a subsequence
QUESTION [5 upvotes]: To establish the relation between weak convergence and characteristic functions, the book I'm studying suggest without proof the following theorem:
For any tight sequence of probability measures, there exists a weakly convergent subsequence.
For the proof the author refers to "a rather difficult theorem", the Helly selection principle.
Can anyone suggest me a more simple and direct proof, at least for unidimensional case?

REPLY [2 votes]: I would refer you to to Billingsley's book Convergence of Probability Measures.  This is an excellent introduction to the area of convergence in distribution.