I am trying to construct a certain bound for the KL divergence between two two numbers. I want to show that $-n D((a+s/\sqrt{n}|| a)\le -\frac{s}{2a(1-a)}+Error(1/\sqrt{n})$ $(a\in(0,1))$
\begin{align*} e^{-nD(a+\frac{s}{\sqrt{n}}||a)}&=\exp\Big(-n\Big(\big(a+\frac{s}{\sqrt{n}}\big)\log\frac{a+\frac{s}{\sqrt{n}}}{a}+\big(1-a-\frac{s}{\sqrt{n}}\big)\log\frac{1-a-\frac{s}{\sqrt{n}}}{1-a}\Big)\Big)\\ &=\exp\Big(\big(-na-{s\sqrt{n}}\log\frac{a+\frac{s}{\sqrt{n}}}{a}+\big(n-a n-{s\sqrt{n}}\big)\log\frac{1-a-\frac{s}{\sqrt{n}}}{1-a}\Big)\Big)\\ \end{align*} I want to use $\log(x)\le x-1$ but I dont see where this is possible.