Real Analysis Answers

 by mathemarical induction on n that

3^n ≥ 2n^2 + 1 for all n ∈ N

Given the function g : R → R defined by

g (x) = {x-1/2x+4 if x̸=−2 and 1/2 x=−2

Find whether or not f is injective and surjective. Find the inverse of f, if it exists.

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

Let a be the supremum of a set of real numbers and let ε > 0 be any real number. there is at least one x ∈ S such that

a−ε<x≤a

where S is the set with the given supremum

Find the following limits

lim( sqrt2n+1− Sqrt2n)

n→∞

lim3n^3 −n+8/(4n(n−1)(n−2))

n→∞

b) Consider the sequence (an) = (−1)^n − 2n

 Explain whether the sequence is monotone increasing or decreasing, whether it is monotone and if limn→∞(an) = −∞

1. Show by using an (ε − N ) argument that lim n->inf 3n^2 −2n+1/2n^2 − 4 =3/2
2. Use an (ε − δ) argument to show that f : R → R be the function defined by

f(x)= {x^2−5x−5 if x≥−1 x^2+x+1 if x<−1}

is continuous at x = −1

f (x) be defined as follows

 f(x) = {x^ 2+8x+15/x+3 ifx<−3

x^2 − 7 if x ≥ −3 }

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

Show that the function f :R -> R defined by f(x) = 2x+ 7 has an inverse by applying the inverse function theorem. Find its inverse also 

Suppose that f :[0, 2] ->R is continuous on [0, 2] and differentiable on (0, 2), and that f (0) = 0, f (1) =1, f (2) =1. (i) Show that there exists c1 belongs to(0,1) such that f'(c1) =1 (ii) Show that there exists c2 belong to (1,2)such that f'(c2)=0(iii) Show that there exists c belongs to(0, 2) such that f'=1/3

An integrable function can have finitely many points of discontinuties. True or false with full explanation

Show that the function f defined by

F(x)=x^3+4x^2+x-6

has a real root in the interval [0,2]

Prove that the complement of every closed set is open.