Math 4640 Final
(1) (a) Find a polynomial p(x) of degree 2 that satises p(x0 ) = y0 ,
p (x0 ) = y0 and p (x1 ) = y1 , where x0 , x1 , y0 , y0 , y1 are given real numbers,
x0 = x1 . Give a formula in the form p(x) = y0 l0 (x) + y0 l1 (x) + y1 l2 (x).
(b) F
Homework 2
Page 30 of Rosenlicht: 6, 7, 10, 11, 13.
6. Show that if a, b, x, y R and if a < x < b and a < y < b, then |y x| < b a.
Solution: Since a < x < b, we have that b < x < a. Adding this inequality to the
inequality for y , we nd
a b < y x < b a.
T
Homework 3
Page 61 of Rosenlicht: 1, 2, 4, 5, 6, 15, 16(a),(c), 17, 22.
1. Verify that the following are metric spaces
(a) all n-tuples of real numbers with
n
|xi yi |
d(x, y ) =
i=1
(b) all bounded innite sequences x = (x1 , x2 , . . .) of elements of R
Homework 4
Page 61 of Rosenlicht: 8, 9, 10, 11, 31, 32, 33, 34, 38.
8. Prove that if the points of a convergent sequence of points in a metric space are reordered,
then the new sequence converges to the same limit.
Solution: Let pn , n N be a sequence in
Homework 5
Page 61 of Rosenlicht: 16(b), 29, 35, 36.
Additional Problems:
1. Suppose that cfw_an , cfw_bn and cfw_cn are sequences in R. Suppose that an L, cn L and
that there exists an integer N such that if n N , then
an bn cn .
Show that bn L.
2. Sup
Homework 6
Page 61 of Rosenlicht: 18, 19, 23.
Additional Problems:
1. Show that lim sn = + if and only if lim(sn ) = .
2. Suppose that there exists a N0 such that if n N0 then sn tn .
(a) Prove that if lim sn = + then lim tn = +;
(b) If lim sn = s and lim
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.,
.,,
-'.
..
-;
i
.
...__...'-_..-..._'..;,4.__c_i
111.18 Prove that lirnnman S Fiancee, with equality holding if and only if the sequence
converges.
Show that highway, 3 Friinnocen.
By the denition, li_m_,Hooa,, 2 li11r11\:_,c,c,inf{an : n 2 N
i
. s
.g
a.
i,
a
'i
.
5
I
.
17. Is the function 3:2 uniformly continuous on R? The function / Why?
Let us show that :32 is not uniformly continuous by showing that V6 > 0 V6 > 0, 33:, y 6 R
such that [2: yl < 5 and [$2 3:2] 2 e. To
Homework 1
Page 30 of Rosenlicht: 6, 7, 10, 11, 13.
Additional Problems:
1. Consider a set X = . Let f and g be functions f : X R and g : X R both having
bounded ranges. Show that
inf cfw_f (x) | x X + inf cfw_g (x) | x X
inf cfw_f (x) + g (x) | x X
i
Homework 1
Page 12 of Rosenlicht: 2, 3, 4, 5, 7, 8.
2. If A is a subset of the set S , show that
(a) (Ac )c = A;
(b) A A = A A = A = A;
(c) A = ;
(d) A = .
Solution:
(a) Let x (Ac )c . Then x Ac , which is the same as x A, and so (Ac )c A. If x A,
/
c
the
Math 4317 : Real Analysis I
Mid-Term Exam 1
25 September 2012
Instructions: Answer all of the problems.
Denitions (2 points each)
1. State the denition of a metric space.
A metric space (X, d) is set X with a function d : X X [0, ) such that
0 d(x, y ) f
Math 4317 : Real Analysis I
Mid-Term Exam 2
1 November 2012
Name:
Instructions: Answer all of the problems.
Denitions (1 point each)
1. For a sequence of real numbers cfw_sn , state the denition of lim sup sn and lim inf sn .
Solution: Let uN = supcfw_sn
Homework 1
Page 12 of Rosenlicht: 2, 3, 4, 5, 7, 8.
Additional Problems:
1. For each k N let Ak be a countable set. Show that
Ak
k=1
is countable. In other words, a countable union of countable sets is countable.
2. Prove that the map f : A B is a bijecti