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Proof: By Induction

(related to Proposition: Inequality between Powers of $2$ and Factorials)

We provide a proof by induction for all natural numbers $n\in\mathbb N.$

• Base case: $n=4$
  • $$16=2^4 < 4!=24.$$
• Induction step $n\to n+1$
  • Assume, $$2^n < n !$$ is correct for an $n\ge 4.$
  • Then $$\begin{align}2^(n+1)&=2\cdot 2 n\nonumber\\ & < 2\cdot n !\quad\text{(by assumption)}\nonumber\\ & < (n+1)\cdot n ! \nonumber\\ & = (n+1) ! \nonumber \end{align}$$
• Thus, the powers of $2$ are smaller than the corresponding factorials for $n\ge 4.$
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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983