ProofWiki:Sandbox

Theorem
Let $x \in \R$ be a real number such that $x > 0$.

Let $\left\langle{ f_n }\right\rangle$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as:
 * $f_n \left({ x }\right) = n \left({ \sqrt[n]{ x } - 1 }\right)$

Then $\forall x \in \R_{>0} : \left\langle{ f_n \left({ x }\right) }\right\rangle$ is strictly decreasing.

Proof
Fix $t \in R_{>0}$.

Then:

Thus:
 * $ n \left({ t^{n + 1} - 1 }\right) > \left({ n + 1 }\right) \left({ t^{n} - 1 }\right)$

Fix $x \in \R_{>0}$.

From Power of Positive Real Number is Positive/Rational Number:
 * $\forall n \in \N : x^{ 1 / \left({ n \left({ n + 1 }\right) }\right) } \in \R_{>0}$

Thus:

Hence the result, by definition of strictly decreasing sequence.