$(3/4) \cdot (8/9) \cdot (15/16) \cdot \ldots$ How to prove that limit is $1/2$ How can one prove $$\frac34 \cdot  \frac89 \cdot \cdots \cdot \frac{n^2-1}{n^2}$$ has a limit $1/2$?
I tried some manipulations with the terms with no success and have just no idea,
Thanks

Rewrite each $1-\frac{1}{k^2}$ as $\frac{(k-1)(k+1)}{k^2}$ and observe the mass cancellations. 
$$\left(\frac{3}{2}\frac{4}{3}\frac{5}{4}\cdots\frac{n}{n-1}\frac{n+1}{n}\right)\cdot\left(\frac{1}{2}\frac{2}{3}\frac{3}{4}\cdots\frac{n-2}{n-1}\frac{n-1}{n}\right)$$
After mass cancellations, pull the $$\frac{n+1}{2}\text{ and }\frac{1}{n}$$
$$\frac{n+1}{2}\cdot\frac{1}{n}$$
Limit of this function tending to infinity $= 1/2$.
This question has been ansewred million times in this site.
Goodluck
