Prove an inequality I want to show that following:
$$\left(\frac{n^2-1}{n^2}\right)^n\sqrt{\frac{n+1}{n-1}}\leq 1; ~~n\geq 2$$ and $n$ is an integer. 
After some simplifications, I got left hand-side as
$$LHS:\left(1-\frac{1}{n}\right)^{n-\frac{1}{2}} \left(1+\frac{1}{n}\right)^{n+\frac{1}{2}}$$
It is clear that the 1st term is less than 1, but I do not have any clue how I can show that multiplication is less than 1.
Can someone give me some hints?  

Let
$$f(x) = \Big(\frac{x^2-1}{x^2}\Big)^x\sqrt{\frac{x+1}{x-1}}$$
Show that $f$ is monotonically increasing for $x\geq 2$ and that $f\rightarrow 1$ as $x\rightarrow\infty$.
Edit: Might be easier to show that $\log f$ is monotonically increasing and approaches zero.
