Lyapunov's CLT Limit Condition

Question

I am trying to show that Lyapunov's condition holds in Lyapunov's CLT, and am left with the trying to show that for some $\delta >0$ $$\underset{n\rightarrow\infty}{lim} \frac{\sum_{i=1}^n w_i^{2+\delta}}{(\sum_{i=1}^n w_i^2)^{2+\delta}}=0$$ Note that $w_i\geq 1,\forall i$. This seems to scream Jensen's inequality, but I still can't get anywhere. Any hints?

Answer 1

Hint: If $\delta <2$ then $$ \sum\limits_{i=1}^{n} w_i^{2+\delta} \leq \max w_i^{\delta} \sum\limits_{i=1}^{n} w_i^{2}.$$ and hence $$ \sum\limits_{i=1}^{n} w_i^{2+\delta} \leq \max w_i^{2} \sum\limits_{i=1}^{n} w_i^{2}$$ which gives $$ \sum\limits_{i=1}^{n} w_i^{2+\delta} \leq (\sum\limits_{i=1}^{n} w_i^{2})^{2}.$$ Now it is easy to finish the proof since $w_i \geq 1$ for all $i$.