Moreover, lets assume there is an input that leads sha-256 to be our target (so the probability of finding a solution isn't 0 at all). Take the shortest input that meets our target, and call it L. Now, brute-force try all inputs up to length L. You know have P(find solution after 2^L tries) = 1. This cannot happen for a memoryless process.
If you pick your inputs randomly, it remains a memoryless process.
