Union of Two Sets: Introductory Analysis Question

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In summary, the given conversation discusses finding the union of intervals with varying limits as n approaches infinity. The first set of intervals, represented by U [2 + 2/n , Pi - 1/n ] for n∈N, has empty intervals for n=1 and n=2, but becomes non-empty when n=4. The second set of intervals, represented by U [1 / (1+n) , 1/n ] for n∈N, has non-empty intervals for n=1, n=2, and n=3, but it is important to consider the behavior of the endpoints as n increases.
  • #1
ShengyaoLiang
23
0
find:

a)

U [2 + 2/n , Pi - 1/n ]
n∈N.

"U" means union. "pi" means 3.1415926535...

b)

U [1 / (1+n) , 1/n ]
n∈N.

thank you so much
 
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  • #2
What's the problem you're having? It would help to take a look at the limits of each "interval" as n goes to infinity, and n=1. Since you're "unioning", you can get a rough idea on how it's going to look and whether one end of the interval becomes open or closed.
 
  • #3
ShengyaoLiang said:
find:

a)

U [2 + 2/n , Pi - 1/n ]
n∈N.

"U" means union. "pi" means 3.1415926535...
Write down some of the intervals: when n= 1, [4, pi-1] is empty because pi-1< 4! when n= 2, [3,pi-1/2] is empty because pi- 1/2< 3! when n= 4, [4/3, pi- 1/3] is non-empty and is precisely that interval. It might be a good idea to draw a few of those on a number line. What is crucially important, as Pseudo Statistic said, is to see what 2+2/n and pi- 1/n converge to as n goes to infinity.

b)

U [1 / (1+n) , 1/n ]
n∈N.

thank you so much
When n= 1 this is [1/2,1], when n= 2, [1/3, 1/2], when n= 3, [1/4, 1/3]. Again, what happens to the endpoints as n goes to infinity? Remember that you are taking a union here.
 
  • #4
thank you very much~~~
 

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