its domain that converges to a point then necessarily converges to It implies that if is a real number (where all are in ) and if is a continuous function whose domain contains and then which (for instance) guarantees that is an adherent point of the set If in addition to what has been assumed, the continuous function is also an increasing or non-decreasing function, then it is even possible to conclude that This may be applied, for instance, to conclude that whenever is a real (or complex) valued function with domain whose sup norm is finite, then for every non-negative real number since the map defined by is a continuous non-decreasing function whose domain always contains and Although this discussion focused on similar conclusions can be reached for with appropriate changes (such as requiring that be non-increasing rather than non-decreasing). Other norms defined in terms of or include the weak space norms (for ), the norm on Lebesgue space and operator norms. Monotone sequences in that converge to (or to ) can also be used to help prove many of the formula given below, since addition and multiplication of real numbers are continuous operations. Arithmetic operations on sets The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets. Throughout, are sets of real numbers. Sum of sets The Minkowski sum of two sets and of real numbers is the set consisting of all possible arithmetic sums of pairs of numbers, one from each set. The infimum and supremum of the Minkowski sum satisfies and Product of sets The multiplication of two sets and of real numbers is defined similarly to their Minkowski sum: If and are nonempty sets of positive real numbers then and similarly for suprema Scalar product of a set The product of a real number and a set of real numbers is the set If then while if then Using and the notation it follows that Multiplicative inverse of a set For any set that does not contain let If is non-empty then where this equation also holds when if the definition is used. This equality may alternatively be written as Moreover, if and only if where if then Duality If one denotes by the partially-ordered set with the opposite order relation; that is, for all declare: then infimum of a subset in equals the supremum of in and vice versa. For