(related to Corollary: Existence of Powers Exceeding Any Positive Constant)

Let \(b > 1\) be a real number and let \(C > 0\) be a constant. We want to show that there exists a natural number \(n\), for which \(b^n > C\).

Set \(x:= b~- 1\). Since \(x > 0\), we can use Bernoulli's inequality and conclude that

\[b^n=(1+x)^n \ge 1 + nx\]

for all \(n\ge 2\). Together with the Archimedean axiom, it follows that there exists a natural number \(n\in\mathbb N\) with \[nx > C - 1.\] Combining both inequalities, we get for this \(n\) that \(b^n > C\).

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