$\ell_{p}$ spaces: Differences between $p$ and $p'$

Question

Suppose $x \in \ell_{p} $ for some $p \in [1, \infty )$. Show that $x \in \ell_{p'}$ for all $p'\geq p$.

So I'm struggling to understand what the difference is between $p$ and $p'$ and how it deals with $\ell$.

Answer 1

The idea is "the larger $p$ is, the easier $\sum|x_n|^p$ converges". The well-known series $\sum(\frac{1}{n})^p$ is a good example.
To prove the statement, recall that since $x\in\ell_p$, we must have $x_n\to 0$. Then for sufficiently large $n$ we have $|x_n|<1$, thus $|x_n|^{p'}\leq|x_n|^p$ for $p'\geq p$. As a conclusion, $x\in\ell_{p'}$.