The objective of this question is part of proving that $\lambda \leqslant \mathrm { e }$. We assume by contradiction that $\lambda > \mathrm { e }$.
Verify that, for all $k$ in $\mathbb { N } , \frac { 1 } { \mathrm { e } } \leqslant \left( \frac { k + 1 } { k + 2 } \right) ^ { k + 1 }$.