Real Analysis Answers

Suppose that 𝛽 > 0 and that ℎ ∈ 𝑅[−𝛽, 𝛽]. 1) If ℎ is even, show that ∫ h(integration limit from [−𝛽, 𝛽]) = 2 ∫ h(integration limit from [0, 𝛽])

Suppose that 𝛽 > 0 and that ℎ ∈ 𝑅[−𝛽, 𝛽]. 1) If ℎ is even, show that t ∫ h(integration limit from [−𝛽, 𝛽]) = 2 ∫ h(integration limit from [0, 𝛽])

Prove or disprove the following statement

‘ Every strictly increasing onto function is invertible'

Prove that

x< log(1/1-x)< x/1-x ; 0<x<1

The limit: limit x→0^+ (xcosecx)^x does not exist

True or false with full explanation

Show that the sequence (an), where an= n/(n^2+4) is monotonic. Is (an ) a Cauchy sequence? Justify your answer

The function: f : [-1,3]→R defined by: f(x)= 3x+1/(x^2+4) is uniformly continuous on [-1,3].

True or false with full explanation

Give an example of a divergent sequence which has two convergent sequences. Justify

your claim.

Which of the following statements are true and which are false? Justify your answers with 

a short proof or a counter-example.

i) If 

x

and 

y

are real numbers such that 

x < y,

then

x^2 < y^2.

The product of two divergent sequences is divergent. True or false? Justify.