This is just Sergei Ivanov's comment finished with Pietro Majer's answer. If all the terms are real and positive then rearrange both $a:=(a_n)$ and $b:=(b_n)$ so they are non-increasing. If $a \neq b$, then by removing the longest initial segment on which they agree we may assume that $a_1 \neq b_1$.

But now

\[ a_1 = \|a\|_\infty=\lim_{p \to \infty} \|a\|_p = \lim_{p \to \infty} \|b\|_p=\|b\|_\infty=b_1, \]

a contradiction.

This is just Sergei Ivanov's comment finished with Pietro Majer's answer. If all the terms are real and positive then rearrange both $a:=(a_n)$ and $b:=(b_n)$ so they are non-increasing. If $a \neq b$, then by removing the longest initial segment on which they agree we may assume that $a_1 \neq b_1$.

But now

$$ a_1 = \|a\|_\infty=\lim_{p \to \infty} \|a\|_p = \lim_{p \to \infty} \|b\|_p=\|b\|_\infty=b_1, $$

a contradiction.

This is just Sergei Ivanov's comment finished with Pietro Majer's answer. If all the terms are real and positive then rearrange both $a:=(a_n)$ and $b:=(b_n)$ so they are non-increasing. By considering the first index at which they disagree we may assume that $a_1 > b_1 \geq b_n$ for all $n$.

But now

\[ a_1 = \lim_{p \to \infty} \|a\|_p = \lim_{p \to \infty} \|b\|_p=b_1, \]

a contradiction.

This is just Sergei Ivanov's comment finished with Pietro Majer's answer. If all the terms are real and positive then rearrange both $a:=(a_n)$ and $b:=(b_n)$ so they are non-increasing. If $a \neq b$, then by removing the longest initial segment on which they agree we may assume that $a_1 \neq b_1$.

But now

\[ a_1 = \|a\|_\infty=\lim_{p \to \infty} \|a\|_p = \lim_{p \to \infty} \|b\|_p=\|b\|_\infty=b_1, \]

a contradiction.