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My understanding of mining is that the reason an award of Bitcoin comes along with the solving of the current block is to give people incentive to actually try to solve the block. My understanding is also that the reason that it is desirable to have people trying to solve blocks is because this is what makes the network work, that is, the more people using their processing power to verify Bitcoin transactions, the better.
What I don't understand is what hashing has to do with actual transaction verification? I sort of vaguely have this idea in my head that because of the difficulty of the SHA256 hashing algorithm, if someone finds a hash that solves a block, that must mean they spent a significant amount of time working to find the hash . . . but what does that have to do with spending a significant amount of time working to verify transactions? What stops people from just hashing but not verifying? Moreover, if I am correct in assuming that hashing serves no other purpose other than to verify that someone indeed put in a fair amount of work for the network, is there really no other better method to prove this? Why is something as stupid as just keeping track of how many transactions a person has verified not going to work?