As sSup s = 0 when s is an empty set of reals, it suffices to show that all elements of s
are at most some nonnegative number a to show that sSup s ≤ a.
As ⨆ i, f i = 0 when the domain of the real-valued function f is empty, it suffices to show
that all values of f are at most some nonnegative number a to show that ⨆ i, f i ≤ a.
As sInf s = 0 when s is an empty set of reals, it suffices to show that all elements of s
are at least some nonpositive number a to show that a ≤ sInf s.
As ⨅ i, f i = 0 when the domain of the real-valued function f is empty, it suffices to show
that all values of f are at least some nonpositive number a to show that a ≤ ⨅ i, f i.
As ⨆ i, f i = 0 when the domain of the real-valued function f is empty,
it suffices to show that all values of f are nonpositive to show that ⨆ i, f i ≤ 0.
As ⨅ i, f i = 0 when the domain of the real-valued function f is empty,
it suffices to show that all values of f are nonnegative to show that 0 ≤ ⨅ i, f i.
As sSup s = 0 when s is a set of reals that's either empty or unbounded above,
it suffices to show that all elements of s are nonnegative to show that 0 ≤ sSup s.
As ⨆ i, f i = 0 when the domain of the real-valued function f is empty or unbounded above,
it suffices to show that all values of f are nonnegative to show that 0 ≤ ⨆ i, f i.
As sInf s = 0 when s is a set of reals that's either empty or unbounded below,
it suffices to show that all elements of s are nonpositive to show that sInf s ≤ 0.
As ⨅ i, f i = 0 when the domain of the real-valued function f is empty or unbounded below,
it suffices to show that all values of f are nonpositive to show that 0 ≤ ⨅ i, f i.