Real Analysis Answers

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.)

Show that {fn}, where fn : [0, 1] → R is the map fn(x) = x

n, is not uniformly convergent.