Proving real inclusion for $1 \leq p \lt r \lt \infty$

Asked Oct 22 '18 at 21:19

Active Oct 22 '18 at 21:38

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How can I show for $1 \leq p \lt r \lt \infty$ that there is the real inclusion

{${x \in \mathbb{R^n}: \left\lVert x \right\rVert_p \leq 1} $} $\subsetneq$ {$x \in \mathbb{R^n}: \left\lVert x \right\rVert_r \leq 1 $}.

I know that there is a real inclusion, if there is $A \subseteq B $ and $A \neq B$, but I don't really know much further..

edited Oct 22 '18 at 21:38

asked Oct 22 '18 at 21:19

To prove that the sets are not equal, you can use the point $x = \left( \frac{1}{\sqrt[r] n}, \ldots , \frac{1}{\sqrt[r] n} \right)\in\mathbb{R}^n$. – MSDG Oct 22 '18 at 21:34
For the inclusion, this post might be useful. – MSDG Oct 22 '18 at 21:44

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