For traditional single-objective optimization problems, there exists a very weak criteria of monotonic algorithm convergence to a solution. This criteria is satisfied even by random sampling with constant memory (save just one best-so-far sample). I introduce an adaptation of this criteria for algorithms attempting to solve so-called ``interactive'' search problems. For such problems, it is non-trivial to construct algorithms that guarantee even this weak monotonic convergence with constant memory. For two types of interactive search problems, algorithms have been proposed that guarantee the weak monotonic convergence, but require unbounded memory in the worst case. I characterize in more detail these memory requirements. Finally, I investigate a third  type of interactive search  problem, for which no algorithms guaranteeing weak convergence have yet been proposed. I speculate that, for this problem type, even with unbounded memory, an algorithm could only provide the guarantee if it explored the space in one of two specific orders (both of which are, unfortunately, prohibitive in practice).