Real Analysis Answers

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)

If 𝜙(x, y) = 0, show that the determinant

|

fxx + λϕxx

fxy + λϕxy

ϕx

fxy + λϕxy

fyy + λϕyy

ϕy

ϕx

ϕy

|

where 𝜆 is Lagrange’s multiplier, is positive, in case the function attains a maximum.

Let 

f

be a differentiable function on 

[,  ]

and 

x [,  ].

Show that, if 

f (x)  0

and 

f (x)  0,

then 

f

must have a local maximum at 

x.

evaluate limit n tends to infinity [n/(1+n^2) + n/(4+n^2) + n/(9+n^2) +.....+n/2n^2

Show that if n is a natural number and α, β are real numbers with β > 0 then there exists a real function f with derivatives of all orders such that: (i) |f^(k)(x)| ≤ β for k ∈ {0, 1, ..., n − 1} and x ∈ (−∞, ∞); (ii) f^(k)(0) = 0 for k ∈ {0, 1, ..., n − 1}; (iii) f ⁽ⁿ⁾(0) = α.

Suppose that y = f(x) : (−∞, ∞) → (−∞, ∞) is infinitely differentiable and has a local minimum at 0. Prove that there exists a disc centered on the y axis which lies above the graph of f and touches the graph at the point (0, f(0)).

] Show that if n is a natural number and α, β are real numbers with β > 0 then there exists a real function f with derivatives of all orders such that: (i) |f^(k)(x)| ≤ β for k ∈ {0, 1, ..., n − 1} and x ∈ (−∞, ∞); (ii) f^(k)(0) = 0 for k ∈ {0, 1, ..., n − 1}; (iii) f ⁽ⁿ⁾(0) = α.