Thread: Proving union and intersection of indexed family of sets

I have to prove that the union for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-infinity,1).
And that the intersection for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-1,0].

I've asked the professor twice now for help and he hasn't been able to explain it at all.

show inclusions in both directions.
for the intersection :what x is between -n and 1/n for all n?

Originally Posted by skippenmydesign

I have to prove that the union for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-infinity,1).
And that the intersection for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-1,0].

I think you are confused by the notation.
Let .
Now consider two examples:

Now let's do an indexed example:


Note how the right and left endpoints are limits.
Do you see how it works?

Ok, I'm not familiar with latex so I can't get this into symbolic form but I'm going to attach a picture of what the question is. I have to prove it using the definition that for two things to be equal they each have to be a subset of the other.

Originally Posted by skippenmydesign

Ok, I'm not familiar with latex so I can't get this into symbolic form but I'm going to attach a picture of what the question is. I have to prove it using the definition that for two things to be equal they each have to be a subset of the other.

Frankly I have no idea why you posted exactly what you posted exactly what I had already posted.


BTW: Why not learn to use LaTeX?

because I need a proof for the answer? not the answer itself? I know the answer but not the proof for it.

Also like I said I know from talking to the professor that the proof needs to use the definition of for things to be equal they have to be subsets of each other. I hope that is clear.

Also I might learn latex sometime but not tonight!

try proving this "in-between" step:

for the first problem-

show that if k < m, that (D_k)U(D_m) = (-m,1/k). what is the smallest k can be, and what is the largest m can be?

for the second problem-

show that for k < m that (D_k)∩(D_m) = (-k,1/m). again: how small can k be, and how large can m be?

(the answer for k in both cases should be "easy". answering for m might take a little thought).

try drawing a picture with k = 3, and m = 4. draw another one with k = 1, and m = 10. what do you notice?