Reference for approximation criterion for convergence in distribution

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I'm studying the article "Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model" (1987, Aizenman, Lebowitz and Ruelle). In the beginning of section 3 of the article, they cite the following result:

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And as a reference, they give Feller's "An Introduction to Probability Theory and Its Applications, Vol II".

Can someone refer me to the location in Feller's book, or another resource, where I can find a statement and proof of this result?

Thank you