Real Analysis

the following statements true or false? Give reasons for your answer.

a) For the function f, defined by f(x) =4x^{2}-4x^{2}- 7x -2,there exists a point

C ∈ ]-1/2,2[ satisfying f′(c) = 0

b) For all even integral values of n,

lim (x+1)^{-n} exists.

x→∞

c) The function f defined by f(x)= [x − 1], (where [x] is the greatest integer

function) is integrable on the interval [2,-4].

d) Every infinite set is an open set.

e) All strictly monotonically decreasing sequences are convergent.

Real Analysis

Q/Prove that 1+x<3^x for all x>0

Q/Prove that there is at least one x€R such that 3^x=3-x

Q/Suppose that lim n_infinity Sn=1 using definition ,prove that lim n to infinity (1+2Sn)=3

Q/(i). State the definition of a Nested Sequence of Sets and write an example. If {In}n€N is a nested sequence of nonempty closed bounded intervals, then Prove that

E=⋂In={x|x € In for all n€N}

contains at least one number. Moreover, if the lengths of these intervals satisfy as , then E contains exactly one number.

Q/(i). State Monotone Convergence Theorem or (MCT).

(ii).Prove that each of the following sequence converges to zero.Sn=(Sin n^4+n+1/n^2+1)/n

Q/prove that x-1/logx is uniformly continuous on(0,1)

Q/Prove that there is at least one x€R such that 3^x=3-x

Q/Suppose that lim n_infinity Sn=1 using definition ,prove that lim n to infinity (1+2Sn)=3

Q/(i). State the definition of a Nested Sequence of Sets and write an example. If {In}n€N is a nested sequence of nonempty closed bounded intervals, then Prove that

E=⋂In={x|x € In for all n€N}

contains at least one number. Moreover, if the lengths of these intervals satisfy as , then E contains exactly one number.

Q/(i). State Monotone Convergence Theorem or (MCT).

(ii).Prove that each of the following sequence converges to zero.Sn=(Sin n^4+n+1/n^2+1)/n

Q/prove that x-1/logx is uniformly continuous on(0,1)

Real Analysis

Consider the set A={n^(-1)^n:n€N} (i). Find maximum and minimum if there exists. (ii). Show that the set is not bounded above. (iii). Show that Inf A =0

Real Analysis

Prove this

Let {xm} be a sequence in Kn, say xm = (x1m,...,xnm). Then

lim m infinity

xm = (x1,..., xn)

with respect to || ||2 if and only if

lim

m infinity

xim = xi

for i = 1,..., n.

Let {xm} be a sequence in Kn, say xm = (x1m,...,xnm). Then

lim m infinity

xm = (x1,..., xn)

with respect to || ||2 if and only if

lim

m infinity

xim = xi

for i = 1,..., n.

Real Analysis

Show,The set U = f(x; y) 2 R2 j x2 + y2 <=1, and x > 0g is open in B(0; 1)

where the norm on R2 is the Euclidean norm.

where the norm on R2 is the Euclidean norm.

Real Analysis

Prove that the series below converges and has a sum<1

1) The summation of 1/(4n-1)(4n+3)as n tends to infinity

2) The summation of1/4n^2-1

1) The summation of 1/(4n-1)(4n+3)as n tends to infinity

2) The summation of1/4n^2-1

Real Analysis

Let f(x)=|x|^3. Show that f'''(0) does not exist

Real Analysis

Show that f:[0,1]_R defined by f(x)=x^2 is uniformly continuous on [0,1]

Real Analysis

Show that the sequence below is monotone,bounded and hence find the limit xn=n+1/n

Real Analysis

Use what you know from analysis on R to come up with a definition of a

Cauchy sequence in V . When would you say V is complete? Is Rn with

|| . ||∞ complete?

Cauchy sequence in V . When would you say V is complete? Is Rn with

|| . ||∞ complete?