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 $\sequence {f_n}$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as:
 * $\map {f_n} x = n \paren {\sqrt [n] x - 1}$

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

Then $\sequence {f_n}$ is uniformly convergent on $K$.

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

From Defining Sequence of Natural Logarithm is Convergent, $\sequence {f_n}$ is pointwise convergent to $\ln$.

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

From Defining Sequence of Natural Logarithm is Strictly Decreasing:
 * $\forall x \in K: \sequence {\map {f_n} x}$ is monotone.

From Dini's Theorem, $\sequence {f_n}$ is uniformly convergent to $\ln$.

Hence the result.