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$r\ge(n-1)/(n-r)$, $rn-r^2\ge n-1$, $rn-r^2-n+1\ge0$, $(r-1)n-(r-1)(r+1)\ge0$, $(r-1)(n-r-1)\ge0$, done.$$ \begin{align*} r &\ge \frac{n-1}{n-r}\\ rn-r^2 &\ge n-1\\ rn-r^2-n+1 &\ge 0\\ (r-1)n-(r-1)(r+1) &\ge 0\\ (r-1)(n-r-1) &\ge 0 \end{align*} $$
$r\ge(n-1)/(n-r)$, $rn-r^2\ge n-1$, $rn-r^2-n+1\ge0$, $(r-1)n-(r-1)(r+1)\ge0$, $(r-1)(n-r-1)\ge0$, done.
$$ \begin{align*} r &\ge \frac{n-1}{n-r}\\ rn-r^2 &\ge n-1\\ rn-r^2-n+1 &\ge 0\\ (r-1)n-(r-1)(r+1) &\ge 0\\ (r-1)(n-r-1) &\ge 0 \end{align*} $$
