Suppose $S_n$ is defined as follows for every positive integer $n\ge 2$: $S_n=\left(1-\tfrac{1}{2^2}\right)\!\left(1-\tfrac{1}{3^2}\right)\cdots\!\left(1-\tfrac{1}{n^2}\right).$ The value of $\displaystyle\lim_{n\to\infty}S_n$ is