CAAM 401/501 HW8 Solutions
Problem 1
Let E be a normed vector space and S a compact set in E .
(a)Let C S and suppose C is closed. Show that C is compact.
(b) Let A E be a closed set and suppose that A and S are not disjoint. Show that A S is compact.
Sol

CAAM 401, 501 Homework 2 Solutions
Fall 2012
Problem 1
Suppose S R, so there is a bijection f : R S . For any a R ,let ga = f (a) S . Consider h(x) = 1 + gx (x).
Obviously, h maps R into R, so h S . Therefore, we can nd a real number c that gc (x) = f (c)

CAAM 401, 501 Homework 3 Solutions
Fall 2012
Problem 1
Consider a set S R, where R is a metric space with the metric d dened by
d(x, y ) = |x y |, x, y R
We consider four denitions of S given below. In each case, determine if the set has any accumulation

1
Problem 1
Lang 0.3.4.
Solution.
(a)
n
nk
n!
(n k )!(n (n k )!
n!
=
(n k )!k !
n!
=
k !(n k )!
n
=
k
=
(b)
n
n
+
k1
k
n!
n!
+
(k 1)!(n (k 1)! k !(n k )!
n!
n!
=
+
[1 2 (k 1)][1 2 (n k + 1)] [1 2 k ][1 2 (n k )]
(n k + 1) n!
k n!
+
=
[1 2 (k 1) k ][1 2 (n

CAAM 401, 501 Homework 2 Solutions
Fall 2012
Problem 1
Suppose S R, so there is a bijection f : R S . For any a R ,let ga = f (a) S . Consider h(x) = 1 + gx (x).
Obviously, h maps R into R, so h S . Therefore, we can nd a real number c that gc (x) = f (c)

CAAM 401, 501 Homework 3 Solutions
Fall 2012
Problem 1
Consider a set S R, where R is a metric space with the metric d dened by
d(x, y ) = |x y |, x, y R
We consider four denitions of S given below. In each case, determine if the set has any accumulation

CAAM 401, 501 Homework 4 Solutions
Fall 2012
Problem 1
Let cfw_an be a sequence in R and let x R
(a) Prove that lim supn an < x implies an < x for n large enough.
(b) Prove that lim supn an > x implies an > x for innitely many n Z+
Solution.
(a) By denit

Problem 1
Consider a sequence cfw_xn of real numbers such that
xn = yn + zn , n Z+
and cfw_yn is a monotone increasing sequence and cfw_zn is a monotone decreasing sequence. Is it true that the sequence cfw_xn
converges? If yes prove it. If not, give

CAAM501FALL2012 HW6 SOLUTION
Problem 1 part 1 (Lang VII.3.1)
Let fn (x) =
xn
for x 0.
1 + xn
(a) Show that fn is bounded.
(b) Show that the sequence cfw_fn converges uniformly on any interval [0, c]
for any number 0 < c < 1.
(c) Show that this sequence c

CAAM 401/501 HW 7 Solutions
Problem 1
Let S be a set in a normed vector space E . We dene the interior S 0 of S by
S 0 := union of all sets V S , such that V is open in E.
We also dene the closure S of S by
S := intersection of all sets S V , such that V

1
Problem 1
Lang 0.3.4.
Solution.
(a)
n
nk
n!
(n k )!(n (n k )!
n!
=
(n k )!k !
n!
=
k !(n k )!
n
=
k
=
(b)
n
n
+
k1
k
n!
n!
+
(k 1)!(n (k 1)! k !(n k )!
n!
n!
=
+
[1 2 (k 1)][1 2 (n k + 1)] [1 2 k ][1 2 (n k )]
(n k + 1) n!
k n!
+
=
[1 2 (k 1) k ][1 2 (n