Proving the Lemma from " A Course in Probability Theory"

[desu87]

I am struggling proving the lemma below.

The lemma is from " A Course in Probability Theory" by Chung, page 244-245.

Let {X_i} (i=1, ..., n) be a sequence of independent random variables, "A" a strictly positive constant (not necessarily the same at each appearance), "?" a constant such that |?|<=A, "P" a probability defined as follows.
?
P= K(n) × 1 / {log(s_n)}^{1??} +?×?_n / (s_n)^3?
where K(n) = 1/ ?{4?(s_n)^2 (1??) loglog(s_n) }

Also, s_n is the standard deviation of?X_i (i=1, ..., n), ?_n the absolute third moment of ?X_i, s_n >=e, s_n?? when n??.

Lemma 1: Suppose that for some ?, 0<?<1, we have

???_n / (s_n)^3 <= A / {log(s_n)}^{1+?} ? (1)

Then for each ?, 0<?<?, we have

??P >= A / {log(s_n)}^{1?(?/2)}??(2)

(end of Lemma 1)

I attempted to prove the lemma by the following, but it doesn't seem to work.

P = K(n) × 1 / {log(s_n)}^{1??} +?×?_n / (s_n)^3
>= K(n) × 1 / {log(s_n)}^{1??} ?|?|×?_n / (s_n)^3
>= K(n) × 1 / {log(s_n)}^{1??} ?|?|×A / {log(s_n)}^{1??}?



In the text book, it is written as follows: "since the first term of P dominates the second term, by (1) since 0<?<?. Hence (2) follow as rather weak consequences."

I couldn't understand the meaning of this statement.

Can somebody help?