Infimun and supremum of empty set

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The infimum of an empty set is defined as infinity and the supremum as negative infinity based on their definitions as the greatest lower bound and least upper bound, respectively. Since there are no elements in the empty set, any real number can be considered an upper bound, leading to the conclusion that the supremum is negative infinity. Conversely, there are no lower bounds for the empty set, resulting in the infimum being defined as infinity. This understanding clarifies the reasoning behind these conventions in mathematical analysis. The discussion highlights the logical foundation for these definitions rather than viewing them as arbitrary conventions.
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Why do we define(by convention) that infimum of an empty set as \infty and supremum as -\infty?
 
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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.
 
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.
 
Thanks Hurkyl and matt grime for your replies. Yes I got it now!
 

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