Derivative of Natural Logarithm Function/Proof 4/Lemma

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

Let $k \in \N$.

Let $J = \closedint {\dfrac 1 k} k$.

Then the sequence of derivatives $\sequence { {f_n}'}_n$ converges uniformly to some real function $g: J \to \R$.

Proof
From Derivative of $n$th Root and Combination Theorem for Sequences:


 * $\forall n \in \N: \forall x \in J : D_x \map {f_n} x = \dfrac {\sqrt [n] x} x$

From Closed Bounded Subset of Real Numbers is Compact, $J$ is compact.

Thus:

In particular, $\sequence { {f_n}'}_n$ is pointwise convergent to a continuous function on $J$.

For $x \in \closedint {\frac 1 k} 1$:

So $\sequence {\map { {f_n}'} x}_n$ is increasing when $x \in \closedint {\dfrac 1 k} 1$.

For $x \in \closedint 1 k$:

So $\sequence {\map { {f_n}'} x}_n$ is decreasing when $x \in \closedint {\dfrac 1 k} 1$.

Thus $\sequence {\map { {f_n}'} x}_n$ is monotone for all $x \in \closedint {\frac 1 k} 1 \cup \closedint 1 k = J$.

From Dini's Theorem, $\sequence { {f_n}'}_n$ converges uniformly to $\dfrac 1 x$ on $J$.

Hence the result.