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Show that $(\sum_{i=1}^{\infty} |x_i|^\beta)^{\frac{\alpha}{\beta}} \leq \sum_{i=1}^{\infty} |x_i|^\alpha$ where $x_i\in \mathbb{R}$ and $0<\alpha<\beta$.

This is a homework question. I'd like to know what material should I look at so that I can solve it by myself. So far, I guess understanding Minkowski inequality might be useful. Any other reading suggestions?

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I cannot really recommend "material". It is about $p$-norms, but the exercise is rather trivial. You should convince yourself that you can reduce it to the case where $0\leq x_j\leq 1$ for all $j$, and from there it is fairly straighforward.

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