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I ran across the following claim in page 5 of this paper, but could not figure out how to prove it. Changing notation a bit for clarity, the claim is:

Let $x_1, \dots, x_n$ be elements of a separable Banach space $(X, \| \cdot \|)$, and suppose $\alpha_1, \dots, \alpha_n$ are scalars with $|\alpha_i| \leq 1$. Then $$\mathbf{E} \biggl\| \sum \epsilon_i \alpha_i x_i \biggr\| \leq \mathbf{E} \biggl\| \sum \epsilon_i x_i \biggr\|$$ where the expectation is taken over the IID Rademacher random variables $\epsilon_1, \dots, \epsilon_n$ (i.e., $\epsilon_i$ takes values $\pm 1$ with equal probability).

I tried assocating every term in the LHS with a term in the RHS that dominates it, but this approach does not seem to work.

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