Derivative of Natural Logarithm Function/Proof 4/Lemma

Theorem
Let $\left\langle{ f_n }\right\rangle_n$ 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 $k \in \N$.

Let $J = \left[{ \frac{1}{k} \,.\,.\, k }\right]$.

Then the sequence of derivatives $\left\langle{ f_n' }\right\rangle_n$ converges uniformly to some real mapping $g : J \to \R$.

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


 * $\forall n \in \N: \forall x \in J : D_x f_n \left({ x }\right) = \dfrac{ \sqrt[n]{x} }{ x }$

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

Thus:

In particular, $\left\langle{ f_n' }\right\rangle_n$ is pointwise convergent to a continuous function on $J$.

For $x \in \left[{ \frac{1}{k} \,.\,.\, 1 }\right]$:

So $\left\langle{ f_n' \left({ x }\right) }\right\rangle_n$ is increasing when $x \in \left[{ \frac{1}{k} \,.\,.\, 1 }\right]$.

For $x \in \left[{ 1 \,.\,.\, k }\right]$:

So $\left\langle{ f_n' \left({ x }\right) }\right\rangle_n$ is decreasing when $x \in \left[{ \frac{1}{k} \,.\,.\, 1 }\right]$.

Thus $\left\langle{ f_n' \left({ x }\right) }\right\rangle_n$ is monotone for all $x \in \left[{ \frac{1}{k} \,.\,.\, 1 }\right] \cup \left[{ 1 \,.\,.\, k }\right] = J$.

From Dini's Theorem, $\left\langle{ f_n' }\right\rangle_n$ converges uniformly to $\frac{1}{x}$ on $J$.

Hence the result.