\(\left(\frac{2^2-1}{2^2}\right)\left(\frac{3^2-1}{3^2}\right)\left(\frac{4^2-1}{4^2}\right)...\left(\frac{\left(n-1\right)^2-1}{\left(n-1\right)^2}\right)\left(\frac{n^2-1}{n^2}\right)\)
=\(\frac{\left(2-1\right)\left(2+1\right)}{2^2}.\frac{\left(3-1\right)\left(3+1\right)}{3^2}.\frac{\left(4-1\right)\left(4+1\right)}{4^2}...\frac{\left(n-2\right)n}{\left(n-1\right)^2}.\frac{\left(n-1\right)\left(n+1\right)}{n^2}\)
=\(\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}...\frac{\left(n-2\right).n}{\left(n-1\right)^2}.\frac{\left(n-1\right)\left(n+1\right)}{n^2}=\frac{1}{2}.\frac{n+1}{n}=\frac{1}{2}+\frac{1}{2n}>\frac{1}{2}\)