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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 that S ⊆ T. Prove that

inf⁡S≥inf⁡T \inf S \geq \inf T

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Solution

Sure, here is the proof:

  1. Since S is a subset of T, every element in S is also in T.

  2. The infimum (inf) of a set is the greatest lower bound of that set.

  3. Because every element of S is also an element of T, any lower bound of T is also a lower bound of S.

  4. Therefore, the greatest lower bound of T (inf T) cannot be greater than the greatest lower bound of S (inf S).

  5. Hence, we have inf S ≥ inf T.

This completes the proof.

This problem has been solved

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