  We investigate the power of weak measurements in the framework of quantum
state discrimination. First, we define and analyze the notion of weak
consecutive measurements. Our main result is a convergence theorem whereby we
demonstrate when and how a set of consecutive weak measurements converges to a
strong measurement. Second, we show that for a small set of consecutive weak
measurements, long before their convergence, one can separate close states
without causing their collapse. We thus demonstrate a tradeoff between the
success probability and the bias of the original vector towards collapse. Next
we use post-selection within the Two-State-Vector Formalism and present the
non-linear expansion of the expectation value of the measurement device's
pointer to distinguish between two predetermined close vectors.
