MAT137Y1 Lecture Notes - Lecture 16: Infimum And Supremum, Empty Set
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Similarly, a is bounded from below if there exists m r, called a lower bound of a, such that x m for every x a. Lower bounds: suppose that a r is a set of real numbers. If m r is an upper bound of a such that m m for every upper bound m of a, then m is called the supremum of a, denoted m = sup a. , inf = : we will only say the supremum or infimum of a set exists if it is a finite real number. Proposition: the supremum or infimum of a set a is unique if it exists. Moreover, if both exist, then inf a sup a. Proof: suppose that m, m are suprema of a. M is an upper bound of a and m is a least upper bound; similarly, m m, so m = m .
