1/n, break the numerator and denominator into separate parts. As n—>infinity, the numerator remains unchanged. So we need only to worry about the denominator. The denominator approaches infinity. 1/∞ = 0.
You may also consider L'Hopital's rule. Take the derivative of the denominator, 1 (a constant) and the derivative is 0. Take the derivative of the denominator, n, and it becomes 1. The derivative of the numerator divided by the derivative of the denominator is 0.