Infinite multiplication

[Siegmund]

If there is some fixed k<1, such that every term in your sequence is less than k, that is sufficient to show convergence to 0. The n_th partial product will be less than k^n and the power series with k<1 converges to 0.

For instance, 0.9 x 0.95 x 0.955 x 0.9555 x 0.95555 ... will converge to zero, because every term is less than 0.96.

Sufficient, but not necessary - the terms can get arbitrarily close to 1 provided they do so very slowly (more slowly than e^(-1/n) approaches 1, I think.)