Real Analysis Answers

Prove that a strictly decreasing function is always one-one

Find the following limit

Lim x tengs to 0 1-cos x^2/x^2 sin x^2

e) Evaluate lim x infty m 1+n^ 2 + m 4+n^ 2 + m 9+n^ 2 +***+ n 2n^ 2 ]

1. Use mathematical induction to show that n! ≥ 2(n-1) for all n ≥ 1.
2. Use (1) and the definition of a Cauchy sequence to show that S_n = ( 1+ 1/2! + 1/3! + ⋯ 1/n!) is Cauchy sequence.

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.