Instructor : Prof. P. Oswald
Graduate TA : Sava Alexandru Undergraduate SA : Rosu Eugenia
Homework 2. Analysis 1
Problem 1.
Prove or disprove the following properties of the supremum:
a) for a nonempty subset of R which is bounded below:
inf A = -sup(-A) where –A={-x | x is in A}
b) for two nonempty subsets of R, call them A and B:
sup(A+B) = sup A + sup B where A+B={x+y | x is in A, y is in B}
c) sup A·B = sup A · sup B where A · B = {xy | x is in A, y is in B}
If any of the above is false give a sufficient condition for it to hold.
Problem 2. (Rudin p. 22 pr. 6) Fix b>1.
a) If m, n, p, q are integers, n>0, q>0, and r = m/n=p/q, prove that:
(bm)1/n = (bp)1/q
Hence it makes sense to define br=(bm)1/n.
b) Prove that br+s = brbs
c) If x is real, define B(x) to be the set of all numbers bt, where t is rational and t ≤ x. Prove
that :
br = sup B(r) if r is rational.
Hence it makes sense to define : bx = sup B(x) for all real x.
d) Prove that bx+y = bxby for all real x and y.
Problem 3.
a) Prove that there is no rational number whose square is 3.
(Hint: In Rudin’s textbook the same problem is solved with 2 instead of 3).
b) Argue that R\Q is infinite.
Problem 4.
Let C be the set of complex numbers (i.e. C = {a+bi | i2 = -1}.
a) Assuming that C is a field, show that C cannot be made an ordered field.
Problem 5.
Let an = 13 + 23+ 33 + … n3 where n runs through the nonzero natural numbers
a) Find a closed form for an. (Hint: compute a few terms. Does it remind you of any other
important sum? Use induction)
b) Let A = {an/n4 | n is in N}. Show that A is bounded and find inf A and sup A.
Bonus Problems:
1) Let {an} be a sequence defined as follows: a1 = 1/2 and an+1=an2 / (an2 – an + 1) for n ≥
1 Prove that for any n ≥ 1 we have that: a1 + a2 + a3 + … +an < 1
(Hint: Try to guess and then prove an equality for 1 – (a1 + a2 + a3 + … +an ) before
proving the inequality. Use induction.)
2) a) Into how many areas do n lines split the Euclidean plane if any 2 of them intersect in a
point, and any 3 have no common point?
b) Into how many space areas do n planes split the Euclidean 3 dimensional space if any
3 of them intersect in exactly one point, and any 4 have no common point?