= Root limit of a finite sum of powers
{title2=$\lim_{p\to\infty}(\sum_{j=1}^m b_j^p)^{1/p}=\max_jb_j$}
For finitely many fixed nonnegative <real numbers> $b_1,\ldots,b_m$, with $m\geq1$,
$$
\boxed{\lim_{p\to\infty}\left(\sum_{j=1}^m b_j^p\right)^{1/p}=\max_{1\leq j\leq m}b_j.}
$$
Let $M$ be that maximum. If $M=0$, every term is zero. Otherwise $M^p\leq\sum b_j^p\leq mM^p$ for $p>0$. Taking roots gives $M\leq(\sum b_j^p)^{1/p}\leq m^{1/p}M$, and $m^{1/p}\to1$. The <squeeze theorem> proves the result, including ties between the largest bases. For example, $(1+a^p+(a^2/2)^p)^{1/p}$ tends to $\max(1,a,a^2/2)$ for $a\geq0$. The finiteness and fixed number of terms supply the upper factor $m$; an infinite <sum> needs further hypotheses.
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