Real Analysis Answers

Use POLYA'S FOUR-STEP PROBLEM SOLVING STRATEGY to solve the problems.

1.  Layla is going to buy 30 multi-vitamin capsules, some P10 and some P18. If she has P340, what is the maximum number of P18 multi-vitamin capsules she can buy? (HINT: Use inequality)
2. Yesterday, Ling got confused about which day of the week it was. Whenever they go to long vacation, he forgets. Johnson said its Friday, but Roger said its Saturday. Ling asked what day is it tomorrow? Roger said its Monday, while Johnson said its Tuesday. Ling asked again, what day has it yesterday? Johnson said Wednesday, while Roger said its Thursday. Johnson and Roger have given one correct answer and two wrong answers. What day is it today? 

show that the inequalities satisfies for all point x,y∈R

d*(x,y) ≤d(x,y)≤√n d*(x,y)

• Let(a↓n) ↓n€B be any sequence. Show that lim↓n-->♾️ a↓n =Liff for every E>0, there exists some N€N such that n>=N implies a↓n N↓e (L)

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