What Are the Interior, Boundary, Closure, and Accumulation Points of These Sets?

[ShengyaoLiang]

a. 1/n + 1/m : m and n are both in N
b. x in irrational #s : x ≤ root 2 ∪ N
c. the straight line L through 2points a and b in R^n.


for part c. i got: intA= empty ; bdA=clA=accA=L Is this correct? how about part a and part b...i am so confused...

 

[HallsofIvy]

As far as the first is concerned I don't know because you didn't say what L is! For (b) you need to know that "between any two irrational numbers, there is at least one rational number". (c) should be easy using the definition of "neighborhood". What does a neighborhood in Rⁿ look like?

 

[SiddharthM]

your c) looks correct.

draw pictures. it will help get rid of the confusion. what textbook are you using for this class?

now for a) and b) determine the sets. Is set A) bounded?

Hint for a's accumulation points, how many points come "near" 2? how about ANY number of the form 1+1/m in between 1 and 2? Fix n=1, let m=1,2,3..., what happens? Fix n as N (N is any fixed integer) and let 1/N +1/m with m=1,2,3... what happens? All these sequences I have suggested are contained in the set A.

for b) do you mean all irrational numbers that are less than the root of 2 and all irrationals that are natural numbers?

edit: werever i say integer, i mean positive integer!

 

[ShengyaoLiang]

don't have a formal texeboot for analysis1, only have a courseware...

thanks a lot.

 

[HallsofIvy]

your c) looks correct.
for b) do you mean all irrational numbers that are less than the root of 2 and all irrationals that are natural numbers?

That would make the second part pretty meaningless!

I would interpret it as b. {x in irrational #s : x ≤ root 2} ∪ N

 

[SiddharthM]

sorry yes, that is actually what i meant to ask.