Homework 3
Assignment: Read pages 47 to 52 in Rudin. Also read the handout on “Sequential Compactness” (which I covered in some fashion last week in class.)
Do problem 12 in Chapter 2. Do problems 1 and 3 in Chapter 3.
Also do the problems below.
1. Find a countable dense subset of R.
2. Give an example of a countable subset of R that is not compact. What
if the set is also bounded—must it be compact? Prove it, or provide a
counterexample.
3. Let E be the set of all numbers in [0, 1] with a decimal expansion that
consists only of the digits 2 and 6. Is E countable? Is E dense in [0, 1]?
Is E compact?
4. Prove, with a careful ϵ−N proof, that the sequence xn = (n2 +n)/(2n2 +
5) in R converges to 1/2.
5. Consider this one “extra credit.” Let A and B be two countably infinite
subsets of the interval (0, 1).
(a) Show that if A and B are dense in (0, 1), then there is a bijective
function f : A → B such that f is strictly increasing on A, i.e.,
for every a1 , a2 ∈ A, f (a1 ) < f (a2 ) whenever whenever a1 < a2 .
(b) Show that such a function does not necessarily exist if A and B
are not dense in (0, 1).
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