Defining Sequence of Natural Logarithm is Uniformly Convergent on Compact Sets

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:
 * $f_n \left({ x }\right) = n \left({ \sqrt[n] x - 1 }\right)$

Let $K \subseteq \R_{>0}$ be compact.

Then $\left\langle{ f_n }\right\rangle$ is uniformly convergent on $K$.

Proof
From Continuity of Root Function and Combination Theorem for Continuous Functions:
 * $\forall n \in \N : f_n \left({x}\right)$ is continuous on $K$

From Defining Sequence of Natural Logarithm is Convergent, $\left\langle{f_n}\right\rangle$ is pointwise convergent to $\ln$.

From Natural Logarithm Function is Continuous, $\ln$ is continuous on $K$.

From Defining Sequence of Natural Logarithm is Strictly Decreasing:
 * $\forall x \in K : \left\langle{ f_n \left({ x }\right) }\right\rangle$ is monotone

From Dini's Theorem, $\left\langle{ f_n }\right\rangle$ is uniformly convergent to $\ln$.

Hence the result.