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} |X_j|/b_n -> 0 in pr. and (S_n - a_n)/b_n converges to a nondegenerate d.f. Then b_n -> 0, 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.