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:
Step 1: Define the problem We are given that S and T are nonempty subsets of R (the set of real numbers) which are bounded below, and that S is a subset of T. We need to prove that inf S (the infimum or greatest lower bound of S) is greater than or equal to inf T (the infimum or greatest lower bound of T).
Step 2: Understand the properties of infimum The infimum of a set is the greatest number that is less than or equal to all numbers in the set. Since S is a subset of T, all elements of S are also elements of T. Therefore, any lower bound of T is also a lower bound of S.
Step 3: Prove the inequality Since inf T is a lower bound of T, and all elements of S are in T, inf T is also a lower bound of S. Therefore, inf S, as the greatest lower bound of S, must be greater than or equal to any other lower bound of S, including inf T.
Therefore, we have proved that inf S ≥ inf T.
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