I am Reading Feller's Introduction to Probability Theory and Its Applications and I found an exercise from Chapter 10 difficult to solve. In this exercise (which is the number 5 in the Intl. Edition) the $X_k$ are mutually independent r.v. and asume values $a_k$ with probability $p_k$, $-a_k$ with probability $p_k$ and $0$ with probability $1-2p_k$. The exercise asks to find conditions on the constants $a_k$ such that the LLN and/or the CLT hold for $\{X_k\}$
In order to solve it I computed the expectation and variance of each $X_k$ so that $\mu_k=0$ and $\sigma^2_k=2a_k^2p_k$. The expectation and variance of the sum of each $X_k$ is $$ m_n = 0 $$ $$ s_n^2 = \sum_{i=1}^{n} 2a_k^2p_k $$ To find a closed-form solution for $s_n^2$ I did the following $$ s_n^2 = \sum_{i=1}^{n} 2a_k^2p_k < \sum_{i=1}^{n} a_k^2 = a_{n+1}^2-a_1^2 $$ So for large $n$, I can approximate $s_n$ as $$ s_n^2 \approx a_{n+1}^2 $$ $$ s_n \approx a_{n+1} $$
Then the LLN holds if $$ \frac{s_n}{n} = \frac{a_{n+1}}{n} \rightarrow \infty $$
And the CLT holds if (Lindeberg's theorem) $$ a_n \leq \epsilon a_{n+1} $$ $$ \frac{a_n}{a_{n+1}} \leq \epsilon $$
This last condition on $a_n$ is weird. It says that the ratio between $a_n$ and its successor should be infinitely small. Moreover, the conditions on the constants for the LLN and the CLT to hold are not in contradiction with each other?
Is this the right way to tackle this problema or did I mess it up?