If a buyer tells you that they only want to purchase a property up to $250,000, you still shouldn't just search the MLS up to that number. In most markets there is negotiation room in list prices. For that reason alone, widen your search by the normal percentage of negotiation room in your market. If sold prices are running 3% below list, divide $250,000 by 0.97 to get the list price of $257,700 that might result in a sale of $250,000.
Conversely, if they gave you a range of $200k to $250k, search from $180,000 or so and higher. What if there is a true bargain out there? You don't want to have to answer their query about this later. At least give them the opportunity to nix the property based on low price or other reasons.