I have to show the inclusion $L^{p}(\mathbb{Z},\mu) \subset L^{q}(\mathbb{Z},\mu)$ with the counting measure. Using 1≤p<q≤$\infty$ and the definition of the $L^{p}/L^{q}$ spaces I have tried to mess around with Hölder's inequality and the integral in the given space (which is just summation) to somehow show the inclusion for a given function but have been unsuccesful until now.
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1$|x|^{q} \leq |x|^{p}$ if $|x| \leq 1$. That is all there is to it. – geetha290krm Jan 04 '23 at 23:30
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2Does this answer your question? Inclusion of $l^p$ space for sequences See not the question but the accepted answer. – Anne Bauval Jan 04 '23 at 23:37
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It does indeed answer my question, but I'm not sure I understand how we got there. First, why does $|x|^{q}≤|x|^{p}$ follow from $|x|≤1$, second, why does $L^{p}(\mathbb{Z},\mu) \subset L^{q}(\mathbb{Z},\mu)$ follow from $|x|^{q}≤|x|^{p}$ ? – hzm Jan 05 '23 at 00:13