Question 2 What is wrong with the argument claiming to prove the following assertion. You should not only find the error, but explain why it is an error providing an example. Assertion: For any two numbers a and 3 define max(a. 3) to be the largest of a er 3. If f and g are continuous at: then so is the function max(f.9) defined by max(f.g)(x) = max(f(r).g(z)). Proof: According to Definition 2.4.1 it must be shown that lim max(f.g)(x) = max(f.g)(=). According to Definition 2.2.1, in order to show this, it must be shown that for every > 0 there is some 6>0 such that max(f.g)(x)-max(f.g)(=) < whenever 0 0 be arbitrary. By the definition of max(f.g)(=)= max(f(=).g(=)) it must be the case that either max(f.g)(=) = f(=) or max(f.g)(=) = g(=) Only the first case will be considered since the argument in the second case is identical. If max (f. 9) (=) = f(=) then there must be some non-empty interval (=-p.=+p) such that max (f.g)(x) = f(x) (1) Juris Steprins 2022. Permission to make digital or hand copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. All other use is prohibited. whenever 0 Question 3 Find the two errors in the following argument purporting to prove the following version of Theorem 2.5.1. As always, an explanation is required. (One of the errors is an error of logic, while the other error is more of a typo.) Assertion: Let p>0. Suppose that for all z such that 0