How do I prove the following inequality involving exponential functions Let $R>2$ be a real number. Then, for any $n\geq 1$, it holds that
$$ \frac{ \exp(n R)+ 1}{\exp( n R) - \exp(\pi R/2)}  \leq \exp(  R/n^2).$$ 
How do I prove this?

The inequality as stated:
$$
   \frac{ \exp(n R)+ 1}{\exp( n R) - \exp(\pi R/2)}  \leq \exp(  R/n^2)
$$
can not be true for all $R>2$ and $n \geq 1$. Indeed, choose $n=2$, then for all $2 < r < r_\ast$, the inequality does not hold, where $r_\ast$ is the unique positive root of 
$$
   e^{2 r}-e^{9 r/4}+\exp\left({\frac{\pi  r}{2}+\frac{r}{4}}\right)+1 
$$
approximately equal $r_\ast \approx 2.118953$.

