Infimun and supremum of empty set

[Edwinkumar]

Why do we define(by convention) that infimum of an empty set as [tex]\infty[/tex] and supremum as [tex]-\infty[/tex]?

 

[Hurkyl]

It's not a convention -- it follows directly from the definition of the supremum as the least upper bound and the infimum as the greatest lower bound.

 

[matt grime]

Remember that we say M is an upper bound for X if for all x in X... so if X is the empty set then this is never true. Now, "false implies true is true", i.e. all possible real numbers are upper bounds for the the empty set.

 

[Edwinkumar]

Thanks Hurkyl and matt grime for your replies. Yes I got it now!