= Simultaneous spectral-radius renorming
{title2=$\|x_j\|_0<r(x_j)+\varepsilon$}
For a finite commuting family in a unital <Banach algebra>, divide each $x_j$ by $r(x_j)+\varepsilon/2$. The <spectral radius formula> makes each normalized element's powers bounded. Commutativity lets every product be reordered into one power of each generator, so the entire generated <semigroup> is bounded. <Renorming a bounded multiplicative semigroup> gives the displayed bounds simultaneously. Applying this to two commuting elements proves submultiplicativity and subadditivity of their <spectral radius>.
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