[tex]\frac{k^2+4}{k+1}\geq \frac{4k+4}{5}\\\\\frac{k^2+4}{k+1}-\frac{4k+4}{5}\geq 0\\\\\frac{k^2+4}{k+1}-\frac{4(k+1)}{5}\geq 0\\\\\frac{5(k^2+4)}{5(k+1)}-\frac{4(k+1)^2}{5(k+1)}\geq 0\\\\\frac{5k^2+20-4(k^2+2k+1)}{5(k+1)}\geq 0\\\\\frac{5k^2+20-4k^2-8k-4}{5(k+1)}\geq 0\\\\\frac{k^2-8k+16}{5(k+1)}\geq 0\\\\\frac{(k-4)^2}{5(k+1)}\geq 0[/tex]
Iloraz liczby nieujemnej i liczby dodatniej jest liczbą nieujemną.
