Assume that the fourth moment of a random variable $X$ defined in $\left(\Omega, \mathcal{A}, P \right)$ satisfies:
$$\mathbb{E} \left( X^4 \right) \leq C \left(\mathbb{E}(X^2) \right)^2, $$
for some constant $C$. For example, this is satisfied if $X\sim N(0,1)$ with $C=3$. Could I deduce from this that
$$ \frac{1}{n} \sum_{i=1}^n X_i^4 \leq C \left( \frac{1}{n} \sum_{i=1}^n X_i^2 \right)^2,$$
for a random sample $\left(X_1, \ldots, X_n \right)$ with probabilty one for large $n$? It seems to me that this assertion follows from the strong law of large numbers but perhaps a more delicate treatment is required. Thank you.