(related to Corollary: Existence of Arbitrarily Small Powers)

Let \(0 < b < 1\) be a real number. We want to show that for every \(\epsilon > 0\) there is natural number \(n\), for which \(b^n < \epsilon\).

Applying the rules for calculation with inequalities (rule 10), we have \(\frac 1b > 1\). According to the corollary about the existence of powers exceeding any positive constant, there exists an natural number \(n\in\mathbb N\), such that \[\left(\frac 1b\right)^n > \frac 1\epsilon.\]

By applying the rules for calculation with inequalities once again, we get \[b^n < \epsilon.\]

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