Real Analysis Answers

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)

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

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.

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.

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

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

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

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

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?

Come up with a definition of uniform convergence for a sequence of functions {fn} on a set A taking values in a normed linear space W over R.
Show that if A = [0, 1], and if {fn} is a sequence of continuous W-valued
functions on [0, 1] which converges uniformly to f : [0, 1] → W, then f is
continuous. (You will have to look at the notes on Lecture 1 that I posted
to learn the definition of continuous functions from a subset C of a normed
space V to a normed space W.)