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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.