2.1.0 Decide which of the following statements are true and which are false. Prove the true
ones and provide a counterexample for the false ones.
(a) If xn converges, then
xn
n
also converges.
(b) If xn does not converge, then
xn
n
does not converge.
(c)
4.1.3 Suppose that
|x| sin 1 , x 6= 0,
x
f (x) =
0,
x = 0.
Show that f (x) is continuous at x = 0 when > 0 and differentiable at x = 0 when > 1.
Proof.
(i) Since | sin x| 1 for all x R, we find, for all x 6= 0,
1
|f (x)| = |x| | sin | |x| .
x
Note that
5.1.1 (a) For f (x) = x3 , compute U (f, P ), L(f, P ) and
P = cfw_0,
R2
0
f (x)dx where
1
, 1, 2.
2
Find out whether the lower sum or upper sum is a better approximation to the integral.
Proof. By the graph analysis, we find
1
1
137
f (1/2) + f (1) + f (
3.1.5 Prove Theorem 3.9: Squeeze Theorem for Functions
Suppose that a R, that I is an open interval which contains a, and that f, g, h are real
functions defined everywhere on I except possibly at a.
(i) If g(x) h(x) f (x) for all x I \ cfw_a, and
lim f (
2.1.0 Decide which of the following statements are true and which are false. Prove the true
ones and provide a counterexample for the false ones.
(a) If xn converges, then
xn
n
also converges.
(b) If xn does not converge, then
xn
n
does not converge.
(c)
4.1.3 Suppose that
|x| sin 1 , x 6= 0,
x
f (x) =
0,
x = 0.
Show that f (x) is continuous at x = 0 when > 0 and differentiable at x = 0 when > 1.
4.1.7 Suppose that f : (0, ) R satisfies f (x) f (y) = f (x/y) for all x, y (0, ) and
f (1) = 0.
(a) Prove t
5.1.1 (a) For f (x) = x3 , compute U (f, P ), L(f, P ) and
P = cfw_0,
R2
0
f (x)dx where
1
, 1, 2.
2
Find out whether the lower sum or upper sum is a better approximation to the integral.
5.1.3 Let E = cfw_1/n : n N. Prove that the function
1, x E,
f (x)
3.1.5 Prove Theorem 3.9: Squeeze Theorem for Functions
Suppose that a R, that I is an open interval which contains a, and that f, g, h are real
functions defined everywhere on I except possibly at a.
(i) If g(x) h(x) f (x) for all x I \ cfw_a, and
lim f (
a
1.2.6 The arithmetic mean of a, b R is A(a, b) = a+b
, and the geometric mean of a, b [0, )
2
is G(a, b) = ab. If 0 a b, prove that a G(a, b) A(a, b) b. Prove that
G(a, b) = A(a, b) if and only if a = b.
1.2.8 Find all values of n N that satisfy the giv
1.2.6 The arithmetic mean of a, b R is A(a, b) = a+b
, and the geometric mean of a, b [0, )
2
is G(a, b) = ab. If 0 a b, prove that a G(a, b) A(a, b) b. Prove that
G(a, b) = A(a, b) if and only if a = b.
Proof. First, since 0 a b, we find
G(a, b) =
ab
aa