Solving Discrete Math Question: Proving ∪n=2∞[0,1 - 1/n] = [0,1)

[bensoa1]

Homework Statement

Show that,
∪_n=2^∞[0,1 - 1/n] = [0,1)

Homework Equations

The Attempt at a Solution

 

[Dick]

Homework Statement

Show that,
∪_n=2^∞[0,1 - 1/n] = [0,1)

Homework Equations

The Attempt at a Solution

Well, explain why the union of all of those intervals contains everything in [0,1] except for 1.

 

[bensoa1]

Well, explain why the union of all of those intervals contains everything in [0,1] except for 1.

This was the plan of action I wanted to take, however, I'm not sure of the appropriate way to do that

 

[Dick]

This was the plan of action I wanted to take, however, I'm not sure of the appropriate way to do that

Start by explaining why it doesn't contain 1. Then continue by explaining why it does contain 99/100. Then extrapolate from there. Think about it.

 

[bensoa1]

Start by explaining why it doesn't contain 1. Then continue by explaining why it does contain 99/100. Then extrapolate from there. Think about it.

My method was to first show that when n = 2 the answer was [0,1). Then I had an arbitrary integer greater than 0, such that ∪_n=2^∞[0,1-1/(k+2)) = [0,1). Would this be sufficient?

 

[Dick]

My method was to first show that when n = 2 the answer was [0,1). Then I had an arbitrary integer greater than 0, such that ∪_n=2^∞[0,1-1/(k+2)) = [0,1). Would this be sufficient?

n goes from 2 to infinity in the union. You can't pick it to be 2. Start by explaining why 1 is not in the union.

 

[bensoa1]

n goes from 2 to infinity in the union. You can't pick it to be 2. Start by explaining why 1 is not in the union.

What is the mathematical proof to use in order to show that it isn't?

 

[bensoa1]

n goes from 2 to infinity in the union. You can't pick it to be 2. Start by explaining why 1 is not in the union.

Okay so I did this, ∪n=2∞[0,1 - 1/n] = [0,1/2), [0,2/3), [0,3/4),...,[0,n-1/n). Since n-1/n < 1, by union properties ∪n=2∞[0,1 - 1/n] = [0,1). Would this suffice?

 

[Dick]

The union is [0,1/2]U[0,2/3]U[0,3/4]U[0,4/5]... Don't you see why 1 isn't in it and any number 0<=x<1 is? Just explain in words.

 

[HallsofIvy]

My method was to first show that when n = 2 the answer was [0,1). Then I had an arbitrary integer greater than 0, such that ∪_n=2^∞[0,1-1/(k+2)) = [0,1). Would this be sufficient?

What do you mean "when n= 2 the answer was [0, 1)"? When n= 2. [0, 1- 1/n]= [0, 1- 1/2]= [0, 1/2]. That is NOT "[0, 1)"!

You need to focus on 1- 1/n. Do you see that there is NO n such that $1\le 1- 1/n$? Why not?

On the other hand, if x is any positive number there exist n such that 1/n< x. Why? Then what can you say about 1- 1/n?

Do you understand that the right hand side is
$$[0, 1/2]\cup[0, 2/3]\cup[0, 3/4]\cup[0, 4/5]\cdot\cdot\cdot$$?