bensoa1 said:Homework Statement
Show that,
∪n=2∞[0,1 - 1/n] = [0,1)
Homework Equations
The Attempt at a Solution
This was the plan of action I wanted to take, however, I'm not sure of the appropriate way to do thatDick said:Well, explain why the union of all of those intervals contains everything in [0,1] except for 1.
bensoa1 said:This was the plan of action I wanted to take, however, I'm not sure of the appropriate way to do that
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 said: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 said: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 is the mathematical proof to use in order to show that it isn't?Dick said: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 said: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 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)"!bensoa1 said: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?
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