Math 414: Analysis I
Homework 3
Due: February 10th, 2014
Name:
The following problems are for additional practice and are not to be turned in: (All
problems come from Basic Analysis, Lebl )
Exercises: 1.3.1, 1.3.2, 1.3.3, 1.3.4
Turn in the following problems.
1. Exercise 1.3.5 from Basic Analysis, Lebl.
2. Exercise 1.3.7 from Basic Analysis, Lebl.
3. Recall the following Characterization Theorem:
If S is a subset of R that contains at least two points and has the property that
• if x, y ∈ S and x < y, then [x, y] ⊂ S,
then S is an interval.
In class, we considered the cases where (i) S was bounded and where (ii) S is
bounded above but not below. Finish the proof of the Characterization Theorem
by proving the cases where (iii) S is bounded below but not above and (iv) S is
neither bounded above nor below.
4. If x, y, z ∈ R and x ≤ z, show that x ≤ y ≤ z if and only if |x − y| + |y − z| = |x − z|.
Interpret this geometrically.
5. If a < x < b and a < y < b, show that |x − y| < b − a. Interpret this geometrically.
6. Let X = Y := {x ∈ R : 0 < x < 1}. Define h : X × Y → R by h(x, y) := 2x + y.
(a) For each x ∈ X, find f (x) := sup {h(x, y) : y ∈ Y }; then find inf {f (x) : x ∈ X}.
(b) For each y ∈ Y , find g(y) := inf {h(x, y) : x ∈ X}; then find sup {g(y) : y ∈ Y }.
Compare your results in (b) with the results found in part (a).
7. Perform the computations in (a) and (b) of the preceding exercise for the function
h : X × Y → R defined by
0 :x<y
h(x, y) =
1 :x≥y
8. (Extra Credit) Let X and Y be nonempty sets and let h : X × Y → R have
bounded range in R. Let F : X → R and G : Y → R be defined by
F (x) := sup {h(x, y) : y ∈ Y } ,
G(y) := sup {h(x, y) : x ∈ X} .
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Math 414: Analysis I
Homework 3
Due: February 10th, 2014
Establish the Principle of the Iterated Suprema:
sup {h(x, y) : x ∈ X, y ∈ Y } = sup {F (x) : x ∈ X} = sup {G(y) : y ∈ Y } .
We sometimes express this in symbols by
sup h(x, y) = sup sup h(x, y) = sup sup h(x, y).
x,y
x
y
y
x
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