Since for sequences $\|a\|_p\to \|a\| _\infty$, the term of maximum absolute value term is determined by the values of $\sum_i a_i^s$ up to a sign, that can be detected looking at $\mathrm{sgn} \sum_i a_i^s$for large odd values of $s$. . Arguing inductively on the powers sum of the remaining terms one concludes. (this assuming all $a_i$ are positive; and similarly if they have a sign).
Since for sequences $\|a\|_p\to \|a\| _\infty$, the term of maximum absolute value is determined by the values of $\sum_i a_i^s$ up to a sign, that can be detected looking at $\mathrm{sgn} \sum_i a_i^s$ for large odd values of $s$. Arguing inductively on the powers sum of the remaining terms one concludes.