Infimun and supremum of empty set

1. May 3, 2009

Edwinkumar

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

Last edited: May 3, 2009
2. May 3, 2009

Hurkyl

Staff Emeritus
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.

3. May 3, 2009

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.

4. May 5, 2009

Edwinkumar

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