Let $\{X_{n}:n\geq1\}$ be a seqeunce of square-integrable independent random variables on $(\Omega,\mathcal{F},\mathbb{P})$ with $\mathbb{E}[X_{n}]=0$ for every $n\geq1$. Set $S_{n}:=\sum_{j=1}^{n}X_{j}$ for each $n\geq1$. Assume that $\sum_{n\geq1}\mathbb{E}[X_{n}^{2}]<\infty$. Prove that, for every $p>1$,
i)$$\big(\mathbb{E}\big[sup_{n\geq1}|S_{n}|^{2p}\big]\big)^{\frac{1}{p}}\leq\frac{p}{p-1}\big(\mathbb{E}\big[|S|^{2p}\big]\big)^{\frac{1}{p}}$$ Without loss of generality assume that $|S|^2 \in L^p$ and you might first consider $min\{\big( sup_n|S_n|^2\big),K\}$ for any fixed $K>0.$
ii) If $S\in L^q$ for some $q\in (2,\infty)$, then $S_n \to S$ also in $L^q$.
Using the said assumptions, it's known that that $S_{n}\to S$ a.s. for some random variable S and due to completeness of $L^2(\Omega,\mathcal{F},\mathbb{P})$, I know that $S \in L^2$ and $S \to S_n$ also in $L^2$. I also can prove that, for every $t>0$,
$$\mathbb{P}\big(sup_{n\geq1} |S_{n}|^2>t\big)\leq\frac{1}{t}\mathbb{E}\big[S^2;sup_{n\geq1}|S_{n}|^2>t\big].$$
I think in order to prove the claim, I can combine the above statement with the following result
$$\mathbb{E}[X^p]=p\int_{0}^{\infty}t^{p-1}\mathbb{P}(X> t)dt = p\int_{0}^{\infty}t^{p-1}\mathbb{P}(X\geq t)dt.$$
My idea is to use Fubini's theorem to get the result, but I don't know how to connect them together.
I'd appreciate any help.
Observe that $$ \mathbb E\left[S^2;\sup_{n\geqslant 1}\left\lvert S_n\right\rvert^2\gt t\right] =\int_{0}^{+\infty}\Pr\left(\left\{S^2\gt s\right\}\cap \left\{\sup_{n\geqslant 1}\left\lvert S_n\right\rvert^2\gt t\right\}\right)\mathrm ds=\int_0^{t/2}+\int_{t/2}^{+\infty}. $$ Bound the first part by $t/2 \Pr\left\{\sup_{n\geqslant 1}\left\lvert S_n\right\rvert^2\gt t\right\}$ and the second one by $\int_{t/2}^{+\infty}\Pr \left\{S^2\gt s\right\}\mathrm ds$. We get $$ t\Pr\left\{\sup_{n\geqslant 1}\left\lvert S_n\right\rvert^2\gt t\right\}\leqslant 2\int_{t/2}^{+\infty}\Pr \left\{S^2\gt s\right\}\mathrm ds. $$