Suppose that S and T are nonempty subsets of R which are bounded below, and thatS ⊆ T . Prove that inf S ≥ inf T
Question
Suppose that S and T are nonempty subsets of R which are bounded below, and thatS ⊆ T . Prove that inf S ≥ inf T
Solution
Sure, here is the proof:
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Since S is a subset of T, every element in S is also in T.
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The infimum (inf) of a set is the greatest lower bound of that set.
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Because every element of S is also an element of T, any lower bound of T is also a lower bound of S.
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Therefore, the greatest lower bound of T (inf T) cannot be greater than the greatest lower bound of S (inf S).
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Hence, we have inf S ≥ inf T.
This completes the proof.
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