Root limit of a finite sum of powers (${\lim }_{p\to \infty }({\sum }_{j=1}^m{b}_j^p{)}^{1/p}={\max }_j{b}_j$)

For finitely many fixed nonnegative real numbers $b_1,\dots ,b_m$, with $m\ge 1$,Let $M$ be that maximum. If $M=0$, every term is zero. Otherwise $M^p\le \sum b_j^p\le m{M}^p$ for $p>0$. Taking roots gives $M\le (\sum b_j^p{)}^{1/p}\le m^{1/p}M$, and $m^{1/p}\to 1$. 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 $a\ge 0$. The finiteness and fixed number of terms supply the upper factor $m$; an infinite sum needs further hypotheses.