Galton–Watson extinction criterion: $q$ is the least root of $f(s)=s$ in $[0,1]$, and $q=1 \iff m\le 1$

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Galton–Watson extinction criterion: $q$ is the least root of $f(s)=s$ in $[0,1]$, and $q=1 \iff m\le 1$ · Prove2Me