Let a R. Prove that, for each n N, there is a rational number rn such that |rn -
ID: 3076736 • Letter: L
Question
Let a R. Prove that, for each n N, there is a rational number rn such that |rn - a| 1/n. For each of the sets below find the supremum and the infimum if they exist. If exist prove that they are indeed the sup and inf. If they do not, justify why they do not. A = {x R: (x + l)(x - 2)(x - 3) 0} B = {-1 -(-1)n/n : n N}. Prove that if b is a lower bound of a set E R and b E then b is the infimum of E. Let A and B be nonempty subsets of R. Prove that if A and B have suprema, then A B has a supremum and sup(A B) = max(sup A,sup B). Prove the following statements by induction 2n + 1 3n for n = 1,2,.... 72n - 1 is a multiple of 6 for n = 1,2,.... 2n n! for n = 4,5,....Explanation / Answer
4)A set A of real numbers (blue balls), a set of upper bounds of A (red diamond and balls), and the smallest such upper bound, that is, the supremum of A (red diamond). In mathematics, given a subset S of a totally or partially ordered set T, the supremum (sup) of S, if it exists, is the least element of T which is greater than or equal to any element of S. Consequently, the supremum is also referred to as the least upper bound (lub or LUB). If the supremum exists, it is unique. If S contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to S (or does not exist). For instance, the negative real numbers do not have a greatest element, and their supremum is 0 (which is not a negative real number). Suprema are often considered for subsets of real numbers, rational numbers, or any other well-known mathematical structure for which it is immediately clear what it means for an element to be "greater-than-or-equal-to" another element. The definition generalizes easily to the more abstract setting of order theory, where one considers arbitrary partially ordered sets. The concept of supremum coincides with the concept of least upper bound, but not with the concepts of minimal upper bound, maximal element, or greatest element. The supremum is in a precise sense dual to the concept of an infimum.
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