Aloha :)
$$a_n=\left(1-\frac12\right)\left(1-\frac13\right)\left(1-\frac14\right)\cdots\left(1-\frac{1}{n-1}\right)\left(1-\frac1n\right)$$$$\phantom{a_n}=\left(\frac22-\frac12\right)\left(\frac33-\frac13\right)\left(\frac44-\frac14\right)\cdots\left(\frac{n-1}{n-1}-\frac{1}{n-1}\right)\left(\frac nn-\frac1n\right)$$$$\phantom{a_n}=\frac{1}{2}\cdot\frac23\cdot\frac34\cdots\frac{n-2}{n-1}\cdot\frac{n-1}{n}=\frac{1}{\pink2}\cdot\frac{\pink2}{\green3}\cdot\frac{\green3}{\blue4}\cdots\frac{\blue{n-2}}{\red{n-1}}\cdot\frac{\red{n-1}}{n}=\frac1n\to0$$