Consider a sequence $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. random variables on a probability space $(\Omega,\mathcal{F}, \mathbb{P})$ with expectation $\mathbb{E}[X_1]=0$ and $\mathbb{E}[X_1^2]<\infty$. Furthermore, let $(c_n)\in l^2$ be a real sequence with $\sum_{k=1}^{\infty}c_k^2<\infty$.
I want to show that $\sup\limits_{n\in\mathbb{N}}\mathbb{E}\left[\left( \sum\limits_{k=1}^nc_k\cdot X_k\right)^{+}\right]<\infty$.
I tried to show that for all $n\in\mathbb{N}$ one has: $\int\limits_{\Omega}\left(\sum\limits_{k=1}^nc_k\cdot X_k\right)^+\leq \sum_{k=1}^{\infty}c_k^2$. But all my attempts failed.
Can you explain to me how to prove it?
Best regards
$$\mathbb{E}\left[\left( \sum\limits_{k=1}^nc_k\cdot X_k\right)^{+}\right]\leq \mathbb{E}\left[\left| \sum\limits_{k=1}^nc_k\cdot X_k\right|\right] \leq \sqrt{\mathbb{E}\left[\left( \sum\limits_{k=1}^nc_k\cdot X_k\right)^2\right]}$$ by Jensen's inequality
and then
$$\mathbb{E}\left[\left( \sum\limits_{k=1}^nc_k\cdot X_k\right)^2\right] = \left(\sum\limits_{k=1}^n c_k^2\right) \mathbb{E}X_1^2$$