A precise description of the nontrivial upper invariant measure for l_e > l c is still an open problem for the basic contact process, which is a self-dual, attractive, but nonreversible Markov process of an interacting particle system. By its selfduality, to identify the invariant measure is equivalent to determining the initial state dependence of the survival probability of the process. A procedure to give rigorous upper bounds for the survival probability is presented based on a lemma given by Harris. Two new bounds are given, improving the simple branching-process bound. In the one-dimensional case, the present procedure can be viewed as a trial to make approximate measures by generalized Markov extensions.