Find limit of this decreasing sequence

$$a_n=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right) \cdots \left(1-\frac{1}{n^2}\right)$$

I have proved that this sequence is decreasing. However I am trying to figure out how to find its limit.

Answer

Hint: Rewrite each $1-\frac{1}{k^2}$ as $\frac{(k-1)(k+1)}{k^2}$ and observe the mass cancellations. It will be useful to do this explicitly for say the product of the first $5$ terms.