Prove that the Least Upper Bound Property holds for ℝ if and only if the Greatest Lower Bound Property holds for ℝ.
To prove that the Least Upper Bound (LUB) Property holds for the set of real numbers ℝ if and only if the Greatest Lower Bound (GLB) Property holds for ℝ, we need to establish two implications:
1. If the LUB Property holds for ℝ, then the GLB Property holds for ℝ.
2. If the GLB Property holds for ℝ, then the LUB Property holds for ℝ.
Let's start with the first implication:
1. If the LUB Property holds for ℝ, then the GLB Property holds for ℝ.
To prove this implication, we'll assume that the LUB Property holds for ℝ, and we need to show that the GLB Property holds as well.
The LUB Property states that every non-empty subset of ℝ that is bounded above has a least upper bound (or supremum). In other words, for any subset S of ℝ that is bounded above, there exists a real number M such that:
i. M is an upper bound of S, meaning that M is greater than or equal to every element of S.
ii. M is the least upper bound of S, meaning that if N is any other upper bound of S, then N is greater than or equal to M.
Now, let's consider the GLB Property. The GLB Property states that every non-empty subset of ℝ that is bounded below has a greatest lower bound (or infimum). In other words, for any subset S of ℝ that is bounded below, there exists a real number m such that:
i. m is a lower bound of S, meaning that m is less than or equal to every element of S.
ii. m is the greatest lower bound of S, meaning that if n is any other lower bound of S, then n is less than or equal to m.
To prove that the GLB Property holds, we can make use of the LUB Property itself. Consider the set A = {-x : x ∈ S}, where S is a non-empty subset of ℝ bounded below.
Since S is bounded below, the set A = {-x : x ∈ S} is bounded above. Furthermore, an upper bound y of A is the negation of a lower bound -x of S.
By the LUB Property, the set A has a least upper bound, say Y. This implies that -Y is a greatest lower bound of S, as -Y is a lower bound of S (since -x ≤ 0 for all x ∈ S) and if n is any other lower bound of S, then -n is an upper bound of A (since -n ≥ 0 for all n ≤ m, and -x ≤ -n for all x ∈ S), thus satisfying the second condition.
Therefore, if the LUB Property holds for ℝ, then the GLB Property holds for ℝ.
Now, let's move on to the second implication:
2. If the GLB Property holds for ℝ, then the LUB Property holds for ℝ.
To prove this implication, we'll assume that the GLB Property holds for ℝ, and we need to show that the LUB Property holds as well.
Let S be a non-empty subset of ℝ that is bounded above. Consider the set B = {-x : x ∈ S}. The set B is non-empty and bounded below, since S is non-empty and bounded above.
By the GLB Property, the set B has a greatest lower bound, say m. This implies that -m is a least upper bound of S, as -m is an upper bound of S (since -x ≤ 0 for all x ∈ S) and if N is any other upper bound of S, then -N is a lower bound of B (since B = {-x : x ∈ S}, we have -x ≥ -N for all x ∈ S), thus satisfying the second condition.
Therefore, if the GLB Property holds for ℝ, then the LUB Property holds for ℝ.
By establishing both implications, we have proven that the Least Upper Bound Property holds for ℝ if and only if the Greatest Lower Bound Property holds for ℝ.