Prove that $h(u)=\frac{2}{u^2}(u-\log(1+u))$ is decreasing on $(-1,\infty)- \{0\}$
This function appears on the proof of Stirling's formula in Principles of Mathematical Analysis, by Walter Rudin.
The classic way to approach the "show a function is decreasing" is to show that its derivative is negative on the given range.
$$h'(u)= -\frac{2}{u^2}+\frac{4}{u^3}\log(1+u)-\frac{2}{u^2(1+u)}$$
And after we factor out the $-\frac{2}{u^2}$ we are left with showing that
$$g(u):= 1-\frac{2}{u}\log(1+u)+\frac{1}{1+u}$$ is positive. We could try showing that $g$ has positive derivative and show that the limit of $g$ as $u$ approaches $-1$ is positive, but this seems like rhe derivative of $g$ will be something much more complicated.