Problem 2 (12 pts.) Let v,z be any real numbers such that vz. Give a direct proof by cases that 2v+zz. Hint: You may use the definition of absolute value on any real number w : - w=w for w0, and - w=w for w0, Hint: We saw direct proof by cases in Week 1's lectures on Proof by Cases, as well as in Tutorial 1. You should first try to understand these proofs before attempting this one. Don't forget to apply the problem solving process by trying the claim out on examples first to get a sense for the result. Grading Notes. While a detailed rubric cannot be provided in advance as it gives away solution details, the following is a general idea of how points are distributed for this problem. If you can at least get part-way, we give partial credit where we can. (10) Correctness. If your proof is not correct, this is where you'll get docked. (2) Regardless of how you formulate your proof, you will need clearly labeled exhaustive cases. (7) Regardless of how you formulate your proof, somewhere you'll need certain facts without which the proof wouldn't work. E.g. if it weren't true that the sum of two integers is integer, would your proof fail? If so, then that is a fact I need to see stated somewhere. (1) The order of these facts makes sense, so that you're not inferring something before you have all the facts to infer it. E.g. you cannot use the fact that the sum of two integers is integer if you don't already know that you have two integers to begin with. The order of how you use these does not have to exactly match those in the sample solutions, but there are orders that will not work and you will lose points if, for example, you use "the difference of ints is int" before you use "the product of ints is int". If you combine some steps (such as "the difference and product of ints is int" or "the product of two non-zero ints is a non-zero int") that is fine. Just don't combine all (see below). (2) Communication. We need to see a mix of notation and intuition, preferably in the "column" format with the notation on the left, and the reasons on the right. If you skip too many steps at once, or we cannot follow your proof, or if your solution is overly wordy or confusing, this is where you'll get docked