Proving Equivalence of (0,1) and [0,1] through Function Mapping

[IntroAnalysis]

Homework Statement

Define : R as follows:

For n element J, n >= 2, f(1/n) = 1/(n - 1)

and for all other x element (0, 1), f(x) = x.

Prove that (0,1) is equivalent to [0,1].

Homework Equations

Equivalent means we must prove that (0,1) is 1-1 and onto [0,1].

The Attempt at a Solution

For n=2, we get f(1/2) = 1/(2 -1) = 1 and as n gets larger, 1/(n - 1) approaches 0. Since n
is an integer, 1/n is rational, so let x represent all irrational numbers in (0, 1).

Additionally, suppose f(x1) = f(x2) and x1 does not = x2, then since f(x) = x, we have
x1 = f(x1) = f(x2) = x2, which is a contradiction. Hence f is 1-1 from (0, 1) into [0, 1].

 

[hunt_mat]

show that f in monotonically increasing in the range [2, infinity)

 

[LCKurtz]

Homework Statement

Define : R as follows:

For n element J, n >= 2, f(1/n) = 1/(n - 1)

and for all other x element (0, 1), f(x) = x.

Prove that (0,1) is equivalent to [0,1].

Homework Equations

Equivalent means we must prove that (0,1) is 1-1 and onto [0,1].

That is a very imprecise and confusing statement. What I'm guessing you mean by equivalent is that there is a function f giving a 1-1 correspondence from (0,1) onto [0,1].

The Attempt at a Solution

For n=2, we get f(1/2) = 1/(2 -1) = 1 and as n gets larger, 1/(n - 1) approaches 0. Since n
is an integer, 1/n is rational, so let x represent all irrational numbers in (0, 1).

Additionally, suppose f(x1) = f(x2) and x1 does not = x2, then since f(x) = x, we have
x1 = f(x1) = f(x2) = x2, which is a contradiction. Hence f is 1-1 from (0, 1) into [0, 1].

That raises several questions, but I will ask you just one. You have defined a function f from (0,1) → [0,1]. For every x in (0,1) except 1/2, 1/3, 1/4, ... you are letting f(x) = x and for the exceptions you are letting f(1/n) = 1/(n-1).

You must show among other things that this map is onto. What x satisfies f(x) = 0?

 

[IntroAnalysis]

show that f in monotonically increasing in the range [2, infinity)

Don't you mean monotonically decreasing?

 

[LCKurtz]

show that f in monotonically increasing in the range [2, infinity)

Don't you mean monotonically decreasing?

Your function maps (0,1) into [0,1]. It doesn't map anything into [2,∞) and it is neither monotone increasing nor decreasing.