Inequality regarding convex combination of random variables

Question

In the appendix of notes on stochastic integration that i am reading, Mazur's Lemma is presented as following https://i.stack.imgur.com/GUyXN.png

I have trouble understanding/proving the following inequality.

Let $(X_n)$ be a sequence of variables bounded in $\mathcal{L}^2$. Define $Z=\sum_{k=n}^{K_n}\lambda_kX_k$ for some convex weights $\lambda_n,...,\lambda_{K_n}$, we obtain $$ \sqrt{E(Z^2)} \leq \sum_{k=n}^{K_n} \lambda_k \sqrt{E(X_k^2)} $$ Can anybody help me prove this inequality? Any help would be appreciated.

Answer 1

Your inequality is equivalent to

$$ E(Z^2) \leq \left(\sum_{k=n}^{K_n} \lambda_k \sqrt{E(X_k^2)}\right)^2 $$

i.e.

$$ \sum_{k=n}^{K_n} \sum_{j=n}^{K_n} \lambda_k \lambda_j E(X_k X_j) \leq \sum_{k=n}^{K_n} \sum_{j=n}^{K_n} \lambda_k \lambda_j \sqrt{E(X_k^2)}\sqrt{E(X_j^2)} $$

Then just remark by Cauchy inequality we have $$E(X_k X_j) \leq \sqrt{E(X_k^2)}\sqrt{E(X_j^2)}$$