ANALYSIS I PRACTICE QUESTIONS (CHAPTERS 1-2)
1.
(a) Let E ⊂ R. Define sup E.
(b) Let
| cos n|
E=
: n ∈ {0, 1, 2, . . . } .
n+1
Find sup E and inf E.
2. Let A ⊂ R be a non-empty set that is bounded above. Show that
sup A = − inf(−A).
3. Let A, B ⊂ R be non-empty sets that are bounded above. We define
A + B = {a + b : a ∈ A, b ∈ B}.
Show that sup(A + B) = sup A + sup B.
4. Prove that for a, b ∈ R, there holds
|a − b| ≥ |a| − |b|.
5. Prove the following inequality for ai ∈ R, i ∈ {1, . . . , n},
X
2
n
n
X
ai ≤ n
a2i .
i=1
i=1
6.
(a) Is the set of irrational numbers countable?
(b) Is the set of all circles in R2 with center p = (x, y) such that x, y ∈ Q
and rational radius r > 0, r ∈ Q countable?
(c) Let N = {0, 1, 2, . . . }. Is the set of all finite subsets of N countable?
(d) Is the set of all subsets of N countable?
7.
(a) Show there is a bijection between {0, 1, 2, 3, . . . } and {1, 2, 3, . . . }.
(b) Show there is a bijection between (0, 1) and [0, 1).
8. True or False.
(a) Open sets cannot be closed.
(b) If U ⊂ R is open and non-empty, then U is uncountable.
(c) There exists a countable subset of R with exactly three limit points.
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2
ANALYSIS I PRACTICE QUESTIONS (CHAPTERS 1-2)
(d) Countable sets are compact.
(e) Let A ⊂ R be arbitrary, and K ⊂ R compact. Then A ∩ K is compact
(f) The intersection of compact sets in R is compact
(g) The closure of a connected set is connected
(h) The interior of a connected set is connected.
9. Determine whether the following subsets of R are open, closed, or compact.
1 1
(a) ∩∞
n=1 (− n , n )
(b) Q
(c) Q ∩ [0, 10]
(d) Z ∩ [0, 10]
10. Define d∞ (x, y) = max{|x1 − y1 |, |x2 − y2 |} on R2 .
(a) Show d∞ is a metric.
(b) Sketch the ball of radius one centered at the origin B1 (0) with respect
to the metric d∞ .
(c) Let U be open in (R2 , d), where d is the usual Euclidean metric d(x, y) =
p
|x1 − y1 |2 + |x2 − y2 |2 . Show that U is open in (R2 , d∞ ).
11. Define d1 (x, y) = |x1 − y1 | + |x2 − y2 | on R2 .
(a) Show d1 is a metric
(b) Show
d(x, y) ≤ d1 (x, y) ≤ 2d(x, y),
p
where d is the usual Euclidean metric d(x, y) = |x1 − y1 |2 + |x2 − y2 |2 .
(c) Show that open sets in (R2 , d) and (R2 , d1 ) are the same.
12. Let A, B, and Aα be subsets of a metric space (X, d).
(a) Show A ∪ B = A ∪ B.
(b) Show ∪Aα ⊂ ∪Aα .
(c) Give an example where ∪Aα 6= ∪Aα .
13. Let (X, d) be a metric space and let K ⊂ X be compact. Prove that if
{Gα }α∈A is an open cover of K, then there exists a number ε > 0 such that
for all x ∈ K, Bε (x) ⊂ Gα for some α ∈ A.
14. Let (X, d) be a metric space. Let K ⊂ X be compact and C be closed.
Show that if K and C are disjoint, then there exists a ε > 0 such that
d(x, y) ≥ ε,
for all x ∈ K and y ∈ C.