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 $\max_{1\leqslant j\leqslant n} \frac{|X_j|}{b_n} \to 0$ in pr. and $(S_n - a_n)/b_n$ converges to a nondegenerate d.f. Then $b_n \to \infty$, $\frac{b_{n+1}}{b_n} \to 1$, and $\frac{a_{n+1} - a_n}{b_n} \to 0$.

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.