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