Fixed point of a compressing function on metric space

Fixed point of a compressing function on metric space See attached file for full problem description with symbols.

Lebesgue Measure and Density : Give an example of a set E such that both E and its complement are dense in R^1. Then show that such a set E can not be closed.

1. Give an example of a set E such that both E and its complement are dense in R^1. Then show that such a set E can not be closed. Note: we are using the "Methods of Real Analysis by Richard R Goldberg" ---

Metric Space and Countable Dense Subset : Prove that if a metric space M is totally bounded, then there is a countable dense subset of M.

2. Prove that if a metric space M is totally bounded, then there is a countable dense subset of M. Note: we are using the "Methods of Real Analysis by Richard R Goldberg

Mapping, Contraction and Fixed-Point Theorem : Let T(x) = x^2 Show that T is a contraction on (0, 1/3] , but that T has no fixed point on this interval. Does this conflict Theorem 6.4? Explain.

(See attached file for full problem description with proper equations) --- 3. Let T(x) = x^2 Show that T is a contraction on (0, 1/3] , but that T has no fixed point on this interval. Does this conflict Theorem 6.4? Explain. Note: We are using the book Methods of Real Analysis by Richard R. Goldberg. This ...continues

Totally Bounded : Let M1 be a totally bounded metric space, and f: M1 --> M2 is uniformly continuous and onto. Show M2 is totally bounded.

6. Let M1 be a totally bounded metric space, and f: M1 --> M2 is uniformly continuous and onto. Show M2 is totally bounded. Note: we are using the "Methods of Real Analysis by Richard R Goldberg" Please see the attached file for the fully formatted problems.

Lebesgue Measure : Why m(Q)=0 and m(In)=2/n?

Please can you explain me with more detail about Lebesgue measure of Q. Why m(Q)=0 and m(In)=2/n? (See attached file for full problem description)

Lebesgue Measures and Integrals : Compute the quantity limit of ( integral from 0 to 1 e^(-x^2/n) dx)

Compute the quantity limit of ( integral from 0 to 1 e^(-x^2/n) dx) ( the integral here is with respect to Lebesgue measure). Make sure that you verify your manipulations by referring to known theorems.

Lebesgue Measures and Integrals : Let a,b be real numbers such that 0 < a < b < infinity. Does the limit [lim of ( integral from a to b of n*sin (x^2/n) dx], n is positive integer exist?

Let a,b be real numbers such that 0 < a < b < infinity. Does the limit lim of ( integral from a to b of n*sin (x^2/n) dx , n is positive integer exist? ( prove or disprove). Find the limit if it exists. Prove all assertions and justify every step. The integral here is with respect of Lebesgue measure.

Lebesgue Measures and Integrals : Let {f_n} be a sequence of nonnegative Lebesgue measurable functions on [0,1]. Suppose that...

Let {f_n} be a sequence of nonnegative Lebesgue measurable functions on [0,1]. Suppose that: (i) f_n -> f in [0,1] and (ii) integral over [0,1] of f_n =< K for all n and some constant K. Then f is in L^1[0,1] and || f||_1 =< K. All integrals are with respect to Lebesgue measure.

Lebesgue Measurable Sets : If the boundary of set omega in R^d has an outer measure zero, then omega is Lebesgue measurable.

If the boundary of set omega in R^d has an outer measure zero, then omega is Lebesgue measurable.