Stirling's Formula/Proof 2/Lemma 3

Lemma
Let $\left \langle {d_n} \right \rangle$ be the sequence defined as:
 * $d_n = \ln \left({n!}\right) - \left({n + \dfrac 1 2}\right) \ln n + n$

Then the sequence:
 * $\left \langle {d_n - \dfrac 1 {12 n}}\right\rangle$

is increasing.

Proof
We have:

Let:
 * $f \left({x}\right) := \dfrac 1 {2 x} \ln \left({\dfrac {1 + x} {1 - x} }\right) - 1$

for $\left\vert{x}\right\vert < 1$.

Then:

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

Thus the sequence:
 * $\left \langle {d_n - \dfrac 1 {12 n}}\right\rangle$

is increasing.