Prove that the set of real numbers of the form e^n,n=
0,=+-1,+-2,... is countable.
Prove that the set of real numbers of the form e^n,n=
0,=+-1,+-2,... is countable.
(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪...
(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪ ... ∪ An is countable. (Hint: Induction.)
(6) Let F be the set of all functions from R to R. Show that |F|
> 2 ℵ0 . (Hint: Find an injective function from P(R) to F.)
(7) Let X = {1, 2, 3, 4}, Y = {5, 6, 7, 8}, T = {∅, {1}, {4},
{1, 4}, {1, 2, 3, 4}}, and S =...
Find the interval of convergence for the power series
∑?=0∞ 8?(?−5)3?+1
Give your answer using interval...
Find the interval of convergence for the power series
∑?=0∞ 8?(?−5)3?+1
Give your answer using interval notation. If you need to use ∞,
type INF. If there is only one point in the
interval of convergence, the interval notation is
[a]. For example, if 0 is the only point in the
interval of convergence, you would answer with [0].
Problem 3 Countable and Uncountable Sets
(a) Show that there are uncountably infinite many real numbers...
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...