Stirling's Formula/Proof 2/Lemma 3

Lemma
Let $\sequence {d_n}$ be the sequence defined as:
 * $d_n = \map \ln {n!} - \paren {n + \dfrac 1 2} \ln n + n$

Then the sequence:
 * $\sequence {d_n - \dfrac 1 {12 n} }$

is increasing.

Proof
We have:

Let:
 * $\map f x := \dfrac 1 {2 x} \map \ln {\dfrac {1 + x} {1 - x} } - 1$

for $\size x < 1$.

Then:

As $-1 < \dfrac 1 {2 n + 1} < 1$ it can be substituted for $x$ in $(1)$:

Thus the sequence:
 * $\sequence {d_n - \dfrac 1 {12 n} }$

is increasing.