Real Analysis Answers

Let k ≥ 0 and f : M → M a k-Lipschitz function. Let ε > 0. Give the largest

number φ > 0, if any, such that ∀x, y ∈ M, d(x, y) < φ implies d(f(x), d(y)) < ε.

Show the series Σ x/(1+n^2.x^2 is uniformly convergenr in [ᾰ,1], for any ᾰ>1.

Show that the series Σ x/(1+n^2.x^2) is uniformly convergent using Weierstrass m test

Suppose that the sequence (sn) converges to s and sn ≤ A for every n. Show that s≤A

Let f: [0,1] to R ne a function defined by

f(x)= 1-x^2

Let P1= { 0,1/2,2/3,1}

P2= { 0,1/4,1/2,3/4,1}

be two partition of the interval [0,1]. Calculate L(P2,f) and U(,P1,f)

Show that the notation {X_i} _i€I implicitly involves the notion of function.

”A real number is rational if and only if it has a periodic decimal expansion.” Define the present usage of the word periodic and prove the statement.

Evaluate,

lim(√n/√n^2+ √n/√(n+3)^2+...√n/√(7n- 3)^2

n→∞

Check, whether the collection G, given by

G' =. { ] 1/(n+2), 1/n [ : n ∈ N}

is an open cover of ]0,1[

Check wether the collection G, given by:

G’ = {]1/(n+2), 1/n[ : n∈N}

is an open cover of ]0,1[.