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Valentin [98]
3 years ago
14

A and B are bounded non-empty subsets of R. For inf(A) to be less than or equal to inf(B), which of the following conditions mus

t be met?
a) For every b in B and epsilon > 0, there exists a in A, such that a < b + epsilon.

b) There exists a in A, and b in B such that a < b.

If neither of these conditions are appropriate, what would be appropriate conditions for inf(A) to be less than or equal to inf(B)?
Mathematics
1 answer:
nirvana33 [79]3 years ago
6 0

Answer:

a) must be met

Step-by-step explanation:

We have two conditions:

a) For every b\in B and \epsilon>0, there exists a\in A, such that a.

b) There exists a\in A and b\in B such that  a.

We will prove that conditon a) is equivalent to inf(A)\leq inf(B)

If a) is not satisfied, then it would exist b\in B and \epsilon >0 such that, for every a\in A, a\geq b+\epsilon. This implies that b+\epsilon is a lower bound for A and in consequence

inf(A)\geq b+\epsilon > b\geq inf(B)

Then, inf(A) \leq inf(B) implies a).

If inf(A) \leq inf(B) is not satisfied then, inf(A) > inf(B) and in consequence exists b\inB such that b-inf(A)=\epsilon >0. Then b-\epsilon=inf(A) and, for every a\in A,

b-\epsilon =inf(A)\leq a.

So, a) is not satisfied.

In conclusion, a) is equivalent to inf(A)\leq inf(B)

Finally, observe that condition b) is not an appropiate condition to determine if inf(A)\leq inf(B) or not. For example:

  • <u>A={0}, B={1}</u>. b) is satisfied and inf(A)=0
  • <u>A={0}. B={-1,1}</u>. b) is satisfied and inf(A)=0>-1=inf(B)

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