E not 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" ---

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

Show that T is a contraction

(See attached file for full problem description with proper equations) --- 3. Let Show that T is a contraction on (0. ,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 theorem 6.4 is in the page 159: "Let be a complete metric space. If T is a contraction on , then there is one and only one point in such that . (This is often stated as "T has precisely one fixed point"). --- Note: we are using the "Methods of Real Analysis by Richard R Goldberg

M is totally bounded

(See attached file for full problem description with proper equations) --- 6. Let be a totally bounded metric space, and is uniformly continuous and onto. Show is totally bounded. Note: we are using the "Methods of Real Analysis by Richard R Goldberg" ---

Lebesgue measure

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)