Real Analysis Answers

1. Use the definition of the limit  to show that the sequence (1 + (−1)^n) is divergent.

1. Use the definition of the limit of a sequence to establish

• lim ( 3n²+1 / 6n²+2 ) = 1/2
• lim ( (n+2)^1/2 - (n)^1/2 ) = 0

(i) Prove by mathemarical induction on n that

3n ≥ 2n2 + 1 for all n ∈ N

(ii) Given the function g : R → R defined by

􏰂 x−1

g (x) = 2x+4 1

2

if x̸=−2 if x=−2

Find whether or not f is injective and surjective.

Find the inverse of f, if it exists.

Prove from first principles (i.e.an ε − δ proof) that f is continuous at the point x = −3.

i) Show by using an (ε − N ) argument that

lim 3n2 −2n+1 = 3

n→∞ 2n2 − 4 2

ii) Use an (ε − δ) argument to show that f : R → R be the function defined by

􏰂x2−5x−5 if x≥−1 f(x)= x2+x+1 if x<−1

is continuous at x = −1.

Find the infimum and supremum in each of the following sets of real numbers: S = {x| − x2 + 6x − 3 > 0

(ii) Let a be the supremum of a set of real numbers and let ε > 0 be any real number.Show that there is at least one x ∈ S such that a−ε<x≤a where S is the set with the given supremum.

If a is a sequence of real numbers, then

△a = (an+1 − an)N

is called the difference sequence of a.

a) Let a be a sequence of real numbers. Find △2a := △(△a)

 b) If a is a convergent sequence of real numbers, prove that △a is a null se- quence.

Let a be a sequence of real numbers and let c ∈ R be a cluster point of a.

Let π:N → N be defined by

π(1) = min{k∈N||ak −c|<1},

π(n+1) = min{k∈N|k>π(n), |ak −c|< 1 } foralln∈N. n+1

(i) Justify the definition of π. (i.e Show that π is well defined.) (ii) Show that π is strictly increasing.

(iii) Prove that the subsequence (aπ(n))N of a converges to c

Let X ⊆ N be an infinite set of natural numbers. Let f :N → X be defined

  by

f(n+1) = min(X−{f(1),f(2),...,f(n)}) forall n∈N.

f(1) = minX,

(i) Justify the definition of f. (i.e Show that f is well defined.)

(ii) Prove that f is a strictly increasing bijection.

Evaluate LaTeX: \int_cF.dr\:\: where LaTeX: F\left(x,y,z\right)=xzi-yzkF(x,y,z)=xzi−yzk and c is the line segment from (3,0,1) to (-1,2,0)