Consider the product of the sequence which has ½ repeated N times. The product is (½)^N; which halves each time. If we want the product to go below x, then we c

Consider the product of the sequence which has ½ repeated N times. The product is (½)^N; which halves each time. If we want the product to go below x, then we can plug in N > log2(1/x). Since for any x > 0 we can find an N, we say lim (½)^∞ = 0. tl;dr Basically, as N goes to infinity, the product gets arbitrarily close to 0.