Prove that $p+\frac{2p-1}{ \left(\frac{1-p}{p}\right)^n -1} \leq \frac{1}{2} - \frac{1}{2n}$, if $0<p<1, n\in \mathbb N$
This is from a recently closed question.
Notice that the fraction on the LHS is well defined. The denominator can be written as
$$\frac{1}{p^n} \left( (1-p)^n - p^n \right)=\frac{1-2p}{p^n} g(p)$$ where $g(p)$ is a polynomial of $p$ taking positive values. When $p=\frac 12$, the LHS becomes $p-\frac{1}{2^n g(\frac 12)}$.
My effort was shown here with variable substitution and taking derivatives. I'd like to know if there's a simpler, elementary proof without calculus. Thanks.