The hierarchy of Liapounov conditions The general setup is an array $(X_{nj} : n \in \mathbb{N}, 1 \leq j \leq k_n)$ of random variables (where of course each $k_n$ is an integer of value at least $1$).
Write $S_{n} := \sum_{j=1}^{k_n} X_{nj}$ for the row sums. 
We assume some independence and normalizations:


*

*The random variables in each row are independent.

*All the random variables have finite second moment.

*$\mathbb{E}(X_{nj})=0 $ for all $n,j$.

*$\mathrm{var}(S_n) = 1$ for all $n$.
In this context, say the array has the property $L(\delta)$ (for $\delta>0$) if 
$$ \sum_{j=1}^{k_n} \mathbb{E} |X_{nj}|^{2+\delta} \longrightarrow 0 \quad \textrm{as} \quad n \rightarrow \infty. $$
The exercise I am stuck with asks me to show that: for $\delta < \delta'$, $L(\delta) \implies L(\delta')$.
Many thanks for your help. 

The implication should be reversed because it is not even sure that $X_{n,j}$ has moments of order $2+\delta'$. 
An idea could be the following: fix a positive $\varepsilon$, write 
$$ \tag{*}\mathbb E \left[|X_{n,j}|^{2+\delta}\right] \leqslant 
\varepsilon^\delta \mathbb E \left[|X_{n,j}|^{2
}\right]+\mathbb E \left[ \left(\frac{|X_{n,j}|}{\varepsilon}\right)^{2+\delta}\mathbf 1\{|X_{n,j}|\gt\varepsilon\}\right].$$
Then notice that 
$$\mathbb E \left[ \left(\frac{|X_{n,j}|}{\varepsilon}\right)^{2+\delta}\mathbf 1\{|X_{n,j}|\gt\varepsilon\}\right] \leqslant 
\mathbb E \left[ \left(\frac{|X_{n,j}|}{\varepsilon}\right)^{2+\delta'}\mathbf 1\{|X_{n,j}|\gt\varepsilon\}\right],$$
plug this into $(*)$, sum and take the $\limsup_{n\to \infty}$.
