Question

Prove that the set of real numbers of the form e^n,n= 0,=+-1,+-2,... is countable.

Answer #1

Prove that the set of constructible numbers is countable

Prove whether or not the set ? is countable.
a. ? = [0, 0.001)
b. ? = ℚ x ℚ
I do not really understand how to prove
S is countable.

1. (a) Let S be a nonempty set of real numbers that is bounded
above. Prove that if u and v are both least upper bounds of S, then
u = v.
(b) Let a > 0 be a real number. Deﬁne S := {1 − a n : n ∈ N}.
Prove that if epsilon > 0, then there is an element x ∈ S such
that x > 1−epsilon.

Show that the set of all rational numbers of the form n/5, where
n is an integer, is countable?

Show that the set of all rational numbers of the form n/5, where
n is an integer, is countable?

Problem 3 Countable and Uncountable Sets
(a) Show that there are uncountably infinite many real numbers
in the interval (0, 1). (Hint: Prove this by contradiction.
Specifically, (i) assume that there are countably infinite real
numbers in (0, 1) and denote them as x1, x2, x3, · · · ; (ii)
express each real number x1 between 0 and 1 in decimal expansion;
(iii) construct a number y whose digits are either 1 or 2. Can you
find a way...

Let E⊆R (R: The set of all real numbers)
Prove that E is sequentially compact if and only if E is
compact

Prove Corollary 4.22: A set of real numbers E is closed and
bounded if and only if every infinite subset of E has a point of
accumulation that belongs to E.
Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real
numbers is closed and bounded if and only if every sequence of
points chosen from the set has a subsequence that converges to a
point that belongs to E.
Must use Theorem 4.21 to prove Corollary 4.22 and there should...

Prove that the set of real numbers has the same cardinality
as:
(a) The set of positive real numbers.
(b) The set of nonnegative real numbers.

Prove that the set of real numbers has the same cardinality
as:
(a) The set of positive real numbers.
(b) The set of non-negative real numbers.

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