  The Paszkiewicz conjecture about a product of positive contractions asserts
that given a decreasing sequence $T_1\ge T_2\ge \dots$ of positive contractions
on a separable infinite-dimensional Hilbert space, the product $S_n=T_n\dots
T_1$ converges strongly. Recently, the first named author verified the
conjecture for certain classes of sequences. In this paper, we prove the
Paszkiewicz conjecture in full generality. Moreover, we show that in some
cases, a generalized version of the Paszkiewicz conjecture also holds.
