HOMEWORK 5 (MATH 3000)
(1) Give an proof that
(a) f (x) = x3 is continuous at x = 0.
x2 + 1
(b) f (x) =
is continuous at x = 1.
x
1
x sin
if x = 0
(c) f (x) =
is continuous at x = 0.
x
0
if x = 0
(2) (a) Using , show that for each n N, the function
x2n
1
HOMEWORK 3 (MATH 3000)
(1) Show the following:
(a) If (xn ) converges to L, then every subsequence (xnk ) of (xn ) converges to L.
Proof. For any
> 0, since (xn ) converges to L, there is N such that
|xn L| <
for all n > N .
Let K N such that nk > N for a
HOMEWORK 3 (MATH 3000)
DUE: WED., FEB. 1
(1) Show the following:
(a) If (xn ) converges to L, then every subsequence (xnk ) of (xn ) converges to L.
(b) If lim x2n = L and lim x2n1 = L, then the sequence (xn ) is convergent to L.
n
n
(c) If there are two
HOMEWORK 2 (MATH 3000)
(1) Find lim xn by any means and then give an -N proof that your limit is correct.
n
cos(n3 )
.
2n
1
if n is odd
3n
.
(b) xn =
1
if n is even
n2
(a) xn =
Proof. Let
> 0 be arbitrary. Set N = maxcfw_
odd, then
1
,
3
|xn 0| =
. Then,
HOMEWORK 1 (MATH 3000)
(1) Prove that there is no rational number whose square is 12.
Proof. If there were a rational number, say r = p/q with p and q relatively prime, satises
r2 = 12. Then one has that p2 = 12q . Since 12 = 3 4, one has that p2 is divid
HOMEWORK 2 (MATH 3000)
DUE: WED., JAN. 25
(1) Find lim xn by any means and then give an -N proof that your limit is correct.
n
cos(n3 )
.
2n
1
if n is odd
3n
.
(b) xn =
1
if n is even
n2
(2) Let
(a) xn =
1
n
if n is odd
1 if n is even
Does lim xn exist? P
HOMEWORK 4 (MATH 3000)
DUE: WED., FEB. 8
(1) For the following sets E . Find the set of interior points E , the set of limit points LE . Is
E open? Is E closed? Is E compact?
(a) E = (0, 1);
(b) E = (0, 1];
(c) E = [0, 1] cfw_2;
(d) E = the set of irratio
HOMEWORK 4 (MATH 3000)
(1) For the following sets E . Find the set of interior points E , the set of limit points LE . Is
E open? Is E closed? Is E compact?
(a) E = (0, 1);
Answer. E = (0, 1), LE = [0, 1]; E is open, not closed, not compact.
(b) E = (0, 1
HOMEWORK 7 (MATH 3000)
DUE: MON., MAR. 12
(1) Let f and g be functions which are uniformly continuous on an interval I . Show that
f + g is also uniformly continuous on I . Is f g always uniformly continuous?
1
1
(2) Show that y = sin( x ) is not uniforml
HOMEWORK 7 (MATH 3000)
(1) Let f and g be functions which are uniformly continuous on an interval I . Show that
f + g is also uniformly continuous on I . Is f g always uniformly continuous?
Proof. Let be arbitrary. Since f and g are uniformly continuous,
HOMEWORK 6 (MATH 3000)
(1) Give an proof that f (x) has a limit at the given c.
(a) f (x) = (x + 2)2 at c = 0.
1
(b) f (x) =
at c = 0.
2+x
x3 1
(c) f (x) =
at c = 1.
x1
Answer. For any > 0, set = maxcfw_1, /4. Then, for any x with 0 < |x 1| < ,
one has
x3
HOMEWORK 6 (MATH 3000)
DUE: FRI., MAR. 1
(1) Give an proof that f (x) has a limit at the given c.
(a) f (x) = (x + 2)2 at c = 0.
1
(b) f (x) =
at c = 0.
2+x
x3 1
(c) f (x) =
at c = 1.
x1
(2) Let f, g : R R be continuous functions. Show that the functions
HOMEWORK 5 (MATH 3000)
DUE: THE END OF WED., FEB. 15
(1) Give an proof that
(a) f (x) = x3 is continuous at x = 0.
x2 + 1
(b) f (x) =
is continuous at x = 1.
x
1
x sin
if x = 0
(c) f (x) =
is continuous at x = 0.
x
0
if x = 0
(2) (a) Using , show that for
HOMEWORK 1 (MATH 3000)
DUE: WED., JAN. 18
(1) Prove that there is no rational number whose square is 12.
(2) Let A be a nonempty subset of R. Recall that the inmum of A, denote by inf A, if inf A
is a lower bound of A and there does not exists with > inf