Real Analysis 4 and 8 Hints Here are some common guideline for proofs that involve sup A a Suppose you are trying to prove sup A x where x is some number One strategy is to show that x is an upper bound of A Then the result follows since sup A is the least upper bound of A To do this Let a A be any element of A and show that x a b Similarly when trying to prove inf A x where x is some number One strategy is to show that x is a lower bound of A Then the result follows since inf A is the greatest lower bound of A To do this Let a A be any element of A and show that x a c If you ever know that sup A y for some number y then sometimes it is helpful to remember that there must exist an a A so that sup A a y Here are some tricks that may come up in N proofs a When trying to solve an L for n junk You can use an intermediate term in between an L and that can be solved directly for n Note This term should have some form of n in the denominator not the numerator e g 1 1 n e n n since e n 1 for n 1 e g 1 1 n2 2n 11 n2 Since n2 n2 2n 11 e g n2 1 1 2n 11 n 1 2 10 e g n2 1 11 By insisting that n 11 we may write n2 1 1 1 1 1 2 11 n n nn 1 n additionally we could have done By insisting that n2 2 11 we may write 1 1 n2 11 n2 n2 1 n2 2 2 2 n2 e g 1 n2 2n 100 we can insist that n be big enough so that 2n then n2 n2 4 and 100 1 1 2 n 2 2n 100 n 4 n2 4 1 n2 2 n2 4 2 n2 HW 1 Let S be a nonempty bounded subset of R Let T 2s s S essentially T is obtained by doubling the elements of S Prove that sup T 2 sup S 2 Use the N definition to prove that an n2 n2 4n 11 converges 3 Assume that an converges to L a Prove that an converges to L b Is the converse true Prove or give a counterexample