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Q. For xn given by the following formulas, establish either the convergence or divergence of the sequence X=(xn)
(a) xn= n/(n+1), (b) xn=(〖(-1)〗^n n)/(n+1) ,
(c) xn=n2/(n+1), (d)xn=(2n2+3)/(n2+1)
Q. Prove that set of reals ℝ is an ordered field.
4.5. Let (X,E) and (Y,K) be two measurable spaces.
We equip the product space X × Y with the σ-algebra σ(Xˆ).

Will a subset of X × Y of the form

A×Y, A∈E

be measurable?

Will a subset of the form

X×B, B∈K be measurable? Is Yˆ a measurable map?
4.4. Let (X1, E1), (X2, E2), (Y1, K1) and (Y2, K2) be measurable spaces.
Let
f1 : (X1,E1) → (Y1,K1), f2 : (X2,E2) → (Y2,K2), be measurable maps.

Construct the map f1 × f2 : X1 × X2 → Y1 × Y2 given by
f1 × f2 (x1, x2) = f1(x1), f2(x2) for all (x1, x2) ∈ X1 × X2
.
Show that f1 × f2 is E1 ⊗E2 −K1 ⊗K2 measurable.
Let f, g be continuous functions on Real, such that f(x) < g(x) ) such that x belongs to Real \ rational.. Show that
f(x) >= g(x) for each x belongs to R.
(a) Let a, b ∈ R with a < b. Define f : [a, b] → R by
f(x) = (
b if x = a,
a if a < x ≤ b.
Use the definition of the Riemann integral to prove that f is integrable on [a, b] and
determine the value of the integral R b
a
f
10. a) Give one example for the following. Justify your choice of examples.
i) A bounded set having no limit point.
ii) A bounded set having infinite number of limit points.
iii) A infinite compact set which is not an interval. (6)
b) Prove that the function f defined by





=
,4 if is irrational
,4 if is rational
( )
x
x
f x
is discontinuous at each real number, using the sequential definition of continuity. (4)
9. a) Using Riemann integration show that ∫
+ =
2
1
2
11 3( x )1 dx . (5)
b) Show that the function
x
f x
1
( ) = is continuous on ]1,0] but not uniformly
continuous
8. a) Check whether the following function has a mean value in the interval ]5,2[




≤ ≤
≤ <
=
3 if 3 5
1 if 2 3
( )
x
x
f x
Does this contradict the mean value theorem? Justify. (3)
b) Find the limit as n → ∞ , of the sum

n n
n
n
n
n
n
4
1
3 1 3 2 3 3
2 2 2 2 2 2
+ +
+
+
+
+
+
L . (4)
5
c) Apply Weierstrass M -test to show that the series ∑ +
4 4
10
n x
converges uniformly
for all x ∈ R .
a) Consider the function f (x) = 2cos x in the interval 





2
,0
π
. Show that
( , ) ( , )
1 2 L P f ≤ L P f and ) ( , ) ( ,
2 1 U P f ≤ U P f where 





=
2
,
3
1
,0
π π
P and






=
2
,
3
,
6
2
,0
π π π
P . (6)
b) Show that the derivative f ′ of the following function f given by





=

=
0 if 0
if 0
1
sin ( )
2
x
x
x
x
f x
exists at x = 0 but f ′ is not continuous at 0 .