= Post-initial invariant interval for a squared-denominator population map
{title2=$4r^2/[b(r+4)^2]\leq N_t\leq r/(4b),\quad t\geq2$}
Starting at $N_1=1/b$ with $r>4$, the next two iterates are the maximum $M=r/(4b)$ and its image $L=4r^2/[b(r+4)^2]$. Since $1/b<L\leq M$ and the <squared-denominator population map> decreases on $[L,M]$, its image there lies in $[L,M]$. Thus the displayed interval is invariant after the initial step, and both bounds are attained. The initial value itself is below $L$, so it must not be included in the lower-bound assertion.
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