prove that the set of real numbers is not a countable set

• Mar 31st 2009, 08:05 AM
rosebud
prove that the set of real numbers is not a countable set
1. Prove that R is equivalent to (0,1) and (0,1) is equivalent to [0,1]. Conclude that R is equivalent to R. Now prove that [0,1] is uncountable. Consider an arbitrary function f: J--> [0,1] and prove that im f is not equal to [0,1]. Thus, there is no function from J onto [0,1], and so [0,1] is uncountable. Suppose that T is a function from J to [0,1].
2. Show that there are sequences {a_n}from infinity when n=1 and {b_n} from infinity when n=1 such that [a_1, b_1] is a subset [0,1], and for each n E J, [a_n+1 , b_n+1] is a subset [a_n, b_n] and T(n) is not an element of [a_n, b_n]
3. Show that {a_n} from infinity when n=1 converges; call the limit A.
4. Prove that A E [a_n, b_n] for each n E J. Conclude that A is not an element im T.
5. Finish the proof that [0,1] is uncountable.
• Mar 31st 2009, 04:01 PM
HallsofIvy
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Originally Posted by rosebud
1. Prove that R is equivalent to (0,1) and (0,1) is equivalent to [0,1]. Conclude that R is equivalent to R.

surely that is not what the problem says! That R is equivalent to [0,1] would follow from this but "R is equivalent to R" is trivial.

To prove that R is equivalent to (0,1), look at the function f(x)= $\frac{1}{x(x-1)}$. To prove that (0,1) is equivalent to [0,1], write the rational numbers in (0, 1) in an ordering $\{r_1, r_2, r_3, ...\}$ (which is possible because the rational numbers are countable) and define f(x)= x if x is irrational, $f(r_1)= 0, f(r_2)= 1, f(r_n)= r_{n-1}$ for n>2.

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Now prove that [0,1] is uncountable. Consider an arbitrary function f: J--> [0,1] and prove that im f is not equal to [0,1]. Thus, there is no function from J onto [0,1], and so [0,1] is uncountable. Suppose that T is a function from J to [0,1].
2. Show that there are sequences {a_n}from infinity when n=1 and {b_n} from infinity when n=1 such that [a_1, b_1] is a subset [0,1], and for each n E J, [a_n+1 , b_n+1] is a subset [a_n, b_n] and T(n) is not an element of [a_n, b_n]
3. Show that {a_n} from infinity when n=1 converges; call the limit A.
4. Prove that A E [a_n, b_n] for each n E J. Conclude that A is not an element im T.
I have no idea what "from infinity" means here.

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5. Finish the proof that [0,1] is uncountable.