Homework
Directions:
1. Do your own work.
2. Write out the problem.
3. Use only one side of the page.
4. One problem per page.
Due 01--09--2026
Page 10: 1 -- 7, 3.
(3751, 5851)
Due 01--09--2026
Page 10: 10, 12, 14, 15, 19, 21, 22, 23. (3751)
Page 10: 10, 13, 14, 15, 19, 21, 22, 23. (5851)
Due 01--14--2026
Page 15: 2, 6, 7, 11 16. (3751)
Page 15: 3, 6, 7, 11 20. (5851)
Due 01--20--2026
Page 22: 4, 8, 11. (3751)
Page 22: 5, 11 13. (5851)
Due 01--26--2026
Page 30: 2, 3, 4, 10, 19, 20. (3751)
Page 30: 2, 3, 8, 11, 19, 20. (5851)
Due 01--27--2026
Page 35: 2, 5, 6, 17. (3751)
Page 35: 3, 5, 7, 17. (5851)
Due 01--28-2026
Page 39: 1 -- 4, 6. (3751)
Page 39: 1 -- 4, 6. (5851)
Due 02--02--2026
1.
Prove that
every nonempty subset of
ℝ
that is bounded below has an infimum.
2.
In the proof of the existence of
y∈ℝ
such that
y2
=2,
show that
sup S
=
inf T.
Page 44: 1, 2, 3, 4. (3751)
Page 44: 1, 2, 4, 7 (Sup only). (5851)
Due 02--06--2026
Page 52: 2, 7, 8, 9. (3751)
Page 52: 3, 7, 8, 9. (5851)
Due 02--09--2026
Page 61: 8, 9, 10, 11, 12. (3751)
Page 61: 8, 9, 10, 11, 12. (5851)
Due 02--16--2026
Page 69: 1, 2, 4, 6, 7, 12. (3751)
Page 69: 1, 2, 6, 7, 12, 18. (5851)
Due 02--16--2026
Page 77: 1, 2, 3, 9, 12. (3751)
Page 77: 1, 2, 3, 9, 12. (5851)
Due 02--20--2026
Page 84: 1, 3, 4, 6, 7, 14. (3751)
Page 84: 1, 3, 6, 7, 14, 15. (5851)
Due 02--23--2026
Page 91: 2, 4, 5, 10. (3751)
Page 91: 2, 4, 5, 10. (5851)
Due 02--25--2026
Page 84: 17, 19. (3751)
Page 84: 14, 19. (5851)
Due 03--11--2026
Page 93: 3, 5, 10. (3751)
Page 93: 5, 6, 10. (5851)
Due 03--13--2026
Page 100: 4, 9, 11, 12. (3751)
Page 100: 4, 9, 12, 18. (5851)
Due 03--16--2026
3.
Suppose that
A⊆ℝ
and
x∈ℝ
is an accumulation point of both
A
and
ℝ∖A.
Show that
x is a boundary point of A.
(3751)
4.
Suppose that
A⊆ℝ
and
x∈ℝ
is a boundary point of A.
Show that x is either an accumulation point of A
or an accumulation point of
ℝ∖A.
(5851)
Due 03--20--2026
Page 332: 5, 6, 7, 8, 14, 16, 18. (3751)
Page 332: 5, 6, 7, 8, 14, 16, 18. (5851)
Due 03--25--2026
Page 336: 1, 2, 3, 5, 7, 8, 15.
(3751)
Page 336: 1, 2, 3, 5, 7, 8, 15.
(5851)
5.
Suppose that
E⊆ℝ
is compact and
A⊆E
is uncountable.
Prove that there exists
x∈E
such that for any open
set
U⊆ℝ
containing
x,
U∩A is uncountable.
Due 03--27--2026
Page 110: 1, 2, 3, 4, 5, 6, 9 ,10, 12.
(3751)
Page 110: 1, 2, 4, 5, 6, 9 ,10, 12, 15.
(5851)
Due 04--01--2026
Page 116: 1, 2, 3, 4, 5, 9 ,10.
(3751)
Page 116: 1, 2, 3, 5, 9 ,10, 12.
(5851)
Due 04--10--2026
Page 129: 1 -- 7, 8, 11.
(3751)
Page 129: 1 -- 7, 8, 12.
(5851)
Due 04--10--2026
Page 133: 2, 7.
(3751)
Page 133: 2, 7, 8.
(5851)
Due 04--13--2026
Page 140: 1, 2, 3, 4, 11, 13, 18.
(3751)
Page 140: 1, 2, 3, 4, 11, 13, 18.
(5851)
Due 04--13--2026
Page 148: 1, 2, 4, 5, 6, 8.
(3751)
Page 148: 1, 2, 4, 5, 6, 8, 14.
(5851)
Due 04--17--2026
Page 160: 8, 9.
(3751)
Page 160: 8, 9.
(5851)