Math 314 Model Test IIIDr. HolmesDecember 8, 20091. Write a formal proof of one of the implications in the biconditional(P ∧ Q) → R ↔ (P → (Q → R))12. Prove the theorem “for any natural number x, S(x) 6= x (or equivalentlyx + 1 6= x)” from the axioms of formal arithmetic. Hint: it is as usualan induction proof. You will see opportunities to use axioms 3 and 4,which we have not used much in problems we have solved. On the realtest, the axioms would be supplied: look in the notes.23. Use the definition of limit to expand the statement limx→4x2= 16 andprove that this is true.34. Prove that the function x5− x + 1 has a zero using the IntermediateValue Theorem (or the special case Theorem 1 of chapter 8). The pointof this exercise is to clearly indicate that you understand the theorem:make sure that you state all the conditions needed to ensure that itapplies.45. Theorem 2 of chapter 8 tells us that the values of any continuous func-tion f on [a, b] are bounded above (there is an M such that for allx ∈ [a, b], f (x) ≤ M).Use Theorem 2 to prove that the values of any continuous functionf on [a, b] are bounded below (there is an m such that for all x ∈[a, b], f(x) ≥ m). Hint: you are going to apply the theorem to adifferent function. Spell out all details, and be sure to explain why theconclusion of Theorem 2 for the different function implies our desiredconclusion.Extra Credit: Prove that there is a c ∈ [a, b] such that for all x ∈ [a, b],f(x) ≤ f(c) (this is like Theorem 2 except the upper bound is actu-ally a value of the function). Hint: explain why the set of all f (x)for x ∈ [a, b] has a least upper bound B: then consider the functiong(x) =1f(x)−B. If there is no c such that f(c) = B, then this func-tion is continuous (why?); this function is unbounded (why?); this is acontradiction (why?)56. State and prove the theorem we would use to evaluate limn→∞an+ bnof the sum of two sequences.Extra Credit: Prove that an increasing sequence whose terms arebounded above converges.67. problem 12, chapter 8. You will get this right