Let $X_1, X_2, \dots, X_n, \dots$ be Rademacher random variables (random signs), i.e. with the distribution $$X_i \sim \left\{ \begin{array}{@{}ll@{}} \ \ \ 1 & \text{with probability} \ \ \ \frac{1}{2},\\ -1 & \text{with probability} \ \ \ \frac{1}{2}. \end{array}\right.$$
Now, let us fix $y\in\ell^2$ - meaning that $y=(y_i)_{i=1}^{\infty}$ and $\sum_{i=1}^{\infty}y_i^2<\infty$. How to prove that $$\Bigg(\mathbb{E}\Bigg|\sum_{i=1}^{\infty} y_iX_i\Bigg|^{p}\Bigg)^{\frac{1}{p}} \ \le \ c\cdot\sqrt{p}\cdot\Bigg(\sum_{i=1}^{\infty}y_i^2\Bigg)^{\frac{1}{2}}?$$ Here $p>1$, and $c$ is a universal constant. I have found it in a paper being called a "standard inequality", but I am stuck on how to obtain this $\sqrt{p}$ factor. I will be glad for any help or insight.