This exercise appears in K.L.Chung's A Course in Probability Theory, Chapter 7.

Ex.7.1-4

Let ${X_j}$ be independent r.v.'s such that $max_{1<=j<=n} \frac{|X_j|}{b_n} -> 0$ in pr. and $(S_n - a_n)/b_n$ converges to a nondegenerate d.f. Then $b_n -> \infty$, $b_{n+1}/b_n -> 1$, and $(a_{n+1} - a_n)/b_n -> 0$.

I found it difficult, and I do not have any idea why this is put in the exercise of CLT.Anyone helps me solve this? Thanks.