I need to prove something that feels rather intuitive but I'm not sure if my reasoning is sufficient.
Prompt: suppose $x \in \ell_p$ for some $p \in [1,\infty)$. Show that $x \in \ell_{p'}$ for all $p' \geq p$
This is pretty intuitive because I know that larger $p$-norms get smaller. This made me think that it was a good idea to take the derivative of a p-norm, but he derivative contains the definition of a $p$-norm multiplied by more terms...
$$ \begin{align*} \cfrac{d}{dp}\|x\|_p &= \cfrac{d}{dp}(|x_1|^p+|x_2|^p+\cdots)^{1/p} \\ &= \frac{d}{dp} e^{\ln(|x_1|^p+|x_2|^p+\cdots)\cdot 1/p}\\ &= e^{\ln(|x_1|^p+|x_2|^p+\cdots)\cdot 1/p} \cdot \frac{d}{dp} \ln(|x_1|^p+|x_2|^p+\cdots){1/p} \end{align*} $$
and I eventually ended up with
$$ \begin{align*} (|x_1|^p+|x_2|^p+\cdots)^{1/p}\cdot \left(\cfrac{\ln(|x_1|^p+|x_2|^p+\cdots)}{p^2} \right) + \cfrac{1}{p}\cdot \frac{\ln(|x_1|)\cdot|x_1|^p+ \ln(|x_2|)\cdot|x_2|^p+ \cdots}{|x_1|^p+|x_2|^p+\cdots} \end{align*} $$
Which I'm not finding super helpful.
Guidance on how to show the prompt would be wonderful.
If $x \in \ell^p$, then $$ \sum_{n=1}^{\infty}|x_n|^p<+\infty.$$ In particular, $$\lim_{n}|x_n|=0.$$
The last equality implies that there is a $n_0$ such that $|x_n| \le 1$ for every $n\ge n_0$.
Therefore, for every $n \ge n_0$ and $p'>p$, $|x_n|^{p'} < |x_n|^p.$ Then, by the comparison test:
$$ \sum_{n=1}^{\infty}|x_n|^{p'}<+\infty$$
and $x \in \ell^{p'}$.