Supremum and infimum of specific sets

AI Thread Summary
The discussion revolves around determining the supremum and infimum of various mathematical sets defined by specific formulas. The participant successfully established that the supremum of set A is infinity by showing that values can be generated that exceed any finite bound. For set E, it was concluded that the supremum is 1 and the infimum is 0, while for set D, the supremum was identified as approximately 0.6666. The conversation also touched on the importance of rigor in proofs, particularly for finite and infinite sets, and included discussions on convergence and limits for set C. Overall, the thread emphasizes the need for careful analysis and mathematical justification in determining supremum and infimum values.
  • #51
Bunny-chan said:
Oh.
4m^2 + n^2 = (2m + n)^2 - 4mn or (2m - n)^2 + 4mn
Right. The 2m-n form is the useful one here.
Write the second equation in post #47 using that.
Can you get from that an upper bound on a?
 
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  • #52
haruspex said:
Right. The 2m-n form is the useful one here.
Write the second equation in post #47 using that.
Can you get from that an upper bound on a?
Would that be \frac{1}{4}?
 
  • #53
Bunny-chan said:
Would that be \frac{1}{4}?
Yes.
 
  • #54
haruspex said:
Yes.
Excellent! And we have 0 as infimum?
 
  • #55
Bunny-chan said:
Excellent! And we have 0 as infimum?
As far as I am aware, there is not universal agreement on whether ℕ includes 0. What have you been taught?
 
  • #56
haruspex said:
As far as I am aware, there is not universal agreement on whether ℕ includes 0. What have you been taught?
Oh, I've forgot that. Particularly my professor doesn't include it either.
 
  • #57
Bunny-chan said:
Oh, I've forgot that. Particularly my professor doesn't include it either.
Maybe it does not matter. If we exclude 0, can you still get 0 as infimum?
 
  • #58
haruspex said:
Maybe it does not matter. If we exclude 0, can you still get 0 as infimum?
It's true that we can never get any value equal to zero, but the infimum doesn't need to be contained in the set, as you said. But I'm not very sure how to verify if 0 is the infimum.
 
  • #59
Bunny-chan said:
It's true that we can never get any value equal to zero, but the infimum doesn't need to be contained in the set, as you said. But I'm not very sure how to verify if 0 is the infimum.
Consider very unequal values of m, n.
 
  • #60
haruspex said:
Consider very unequal values of m, n.
The values indeed get more and more close to zero.
 
  • #61
Bunny-chan said:
The values indeed get more and more close to zero.
Right.
 
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  • #62
haruspex said:
Right.
Thank you for this loooong aid!
 

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