See Durrett (2nd ed) section 2.4b for an equivalent formulation and a proof using characteristic functions. That proof leans on the continuity theorem for characteristic functions, (3.4) on page 99, which in turn relies on the Helly selection theorem (2.5) on page 88. The present approach, due to Lindeberg, is more elementary in that it does not require these tools. But note that the basic idea...