- To prove that there exists an such that holds, it suffices to give a specific and then prove that is true for that . Such a proof usually starts "let ", and then goes on to prove that holds for the given . is sometimes referred to as a witness for .
- If you know there exists some satisfying , you can use it in a proof by treating as an arbitrary value. is arbitrary because the only thing you know about is that it exists, not what its value is.
- To disprove that there exists an logical negation of "there exists an such that " is "for all x, not P". satisfying , you must disprove for an arbitrary . Put another way, the
Often, in a proof, it is not immediately obvious what the witness should be. Finding one often involves solving some equations or combining some known values.
One nice technique for finding a witness is to simply leave a blank space for the value of and continue on with your proof of . As you go, you may need to satisfy certain properties (for example, maybe you need at one point, and later you need ). You can make a "wishlist" on the side of your proof, reminding you of all the properties you want to satisfy. Once you've completed your proof, you can go back and find a specific value of (say, ) that satisfies all of your wishes.