Assignment 4
AMATH/PMATH 331
Due on 14 March 2016
1. Show that cfw_|xk | converges in R if cfw_xk converges in a normed vector space V .
2. Show that every convergent sequence in a normed vector space must be a Cauchy sequence.
3. Let f be a continuous f

SOME COUNTABILITY RESULTS FOR NAPIER,
CO-ALMOST SURELY n-DIMENSIONAL, SYMMETRIC
GRAPHS
Q. ZHOU
Abstract. Assume we are given a generic category Bv . It has long been
known that Ramanujans condition is satisfied [12]. We show that T
2. O. W. Kolmogorovs d

Solution 4: AMATH/PMATH 331, 2014
1. Assume that cfw_xk converges in V . Then cfw_xk is a Cauchy sequence. Thus for any > 0, N > 0 such that
for all k, m > N , kxk xm k < . By the triangle inequality, we have
kxk k kxk xm k + |xm k, kxm k kxm xk k + kxk

Solution 2: AMATH/PMATH 331, 2014
1. Note that
n
3n + 3n = 3 2.
Since n 2 1 as n , we get using the squeeze theorem n 2n + 3n 3 as n .
2. Note that a1 = 0, a2 = 5 + 2a1 =
Assume an
an+1 4. Then 2an 2an+1 8, and
5 4.
5 + 2an 5 + 2an+1 13. Hence 5 + 2an

Assignment #1
AMATH/PMATH 331
Due on Monday, 18 January 2016
1. Is each of the following sentences true or false? Just say TRUE or FALSE.
(a) If a b and b a, then a = b.
(b) If a b and c d, then a c b d.
(c) If S = cfw_1, 2, 3, then there exists an x S su

Solution 5: AMATH/PMATH 331, 2014
1. D is a subset of the normed vector space C[0, 2] with the uniform norm. kfk k 1. Thus the set D is
bounded. As proved in Problem 6 of Assignment 4, the sequence cfw_fk converges pointwise to the following
function:
0

Assignment #2
AMATH/PMATH 331
Due on 1 February 2016
1. Show the sequence cfw_ n 2n + 3n converges and find its limit.
2. Show the sequence cfw_xn defined by a1 = 0; an+1 = 5 + 2an converges and find its limit.
3. If a sequence cfw_xn satisfies that xn

Assignment 3
AMATH/PMATH 331
Due on 12 February 2016
1. Using an - argument, show that f (x) =
1
x
is continuous at every point x0 > 0.
2. Let f : R R be the function defined by
x if x is irrational
f (x) =
0 if x is rational.
Prove that f is continuous a

Assignment 5
AMATH/PMATH 331
Due on March 28, 2016
1. Let D = cfw_fk (x), k = 1, 2, , where fk (x) =
but not compact.
2. Let D = cfw_fk (x) =
sin kx
1+k , k
xk
1+xk
for x [0, 2]. Show that the set D is bounded, closed,
= 0, 1, , x [0, 1]. Show that D is c

ON THE EXISTENCE OF HYPERBOLIC, ULTRA-ADDITIVE, ARTINIAN PRIMES
H. MARTINEZ
In [5], it is shown that
Abstract. Suppose we are given a factor .
V,y > F (i, 1) .
We show that
(, )
=
X
log
1
.
Hence every student is aware that every point is almost sure