Assignment 2.
1. Prove that if A and B are non-empty subsets of the real line, both
of which are bounded above, then A+B is bounded above, where A+B is the set of all
points a+b such that a is in A and b is in B. Also, prove that
sup(A+B)=sup(A) +sup(B).
2.ntbhi(not to be handed in) Do the same if A and B are subsets of the
positive numbers, which are bounded above and show AB is bounded
above, with sup(AB)=sup(A)sup(B).
3. Prove that inf{1/n : n a natural numbers}=0. Hints available on
request.
4.Prove that for each real number x, there exists a unique integer n
such that n is less than or equal to x and x is less than n+1.
5. Give an example of irrationals whose supremum is rational. Explain
completely.
More to come.
Assignment 3.
43/2(ntbhi),3-5,17
(ntbhi)Let {a_n} be a sequence of real numbers. Prove that {a_n} is unbounded if and only if there exists a subsequence {a_{n_k}} such that |a_{n_k}}|>k.
Assignment 4.
43/7, 9, 11(you can use what I showed in class, if you include the
proof), 22
There may be more.
Assignment 5 (due Tues., 10/28).
44/13-16, 29
Assignment 6 (due Thurs., 11/06).
78/1-3