Yes. Easiest example would be just the product of a_n, n=1,2,3,..., with a_n=same constant in (0,1). Then all the partial products will be positive, but the l

Yes. Easiest example would be just the product of a_n, n=1,2,3,..., with a_n=same constant in (0,1). Then all the partial products will be positive, but the limit is clearly 0.