Friday, November 02, 2007
A simple advanced calculus/real analysis exercise
Okay, the following is nothing special (upper division undergrad stuff) but it is instructive:
The supremum of a nonempty set is an upper bound (not necessarily contained in the set) such that every element of the set is less than or equal to it and among all upper bounds of the set it is the least. For example, the supremum of the interval [1,5) is 5.
Now, say a set A is contained in a set B (i.e., A is a subset of B). That means if a is an element of A then a is an element of B. By definition, a is less than or equal to the supremum of A. Also, since a is an element of B, then a is less than or equal to the supremum of B. That means the supremum of B is an upper bound for the set A. (The element a is arbitrary.) However, the supremum of A is the least among the ubber bounds, so it must be that the supremum of A is less than or equal to the supremum of B.
Quod erat demonstrandum
Incidentally, another, more descriptive, name for supremum is least upper bound.

The supremum of a nonempty set is an upper bound (not necessarily contained in the set) such that every element of the set is less than or equal to it and among all upper bounds of the set it is the least. For example, the supremum of the interval [1,5) is 5.
Now, say a set A is contained in a set B (i.e., A is a subset of B). That means if a is an element of A then a is an element of B. By definition, a is less than or equal to the supremum of A. Also, since a is an element of B, then a is less than or equal to the supremum of B. That means the supremum of B is an upper bound for the set A. (The element a is arbitrary.) However, the supremum of A is the least among the ubber bounds, so it must be that the supremum of A is less than or equal to the supremum of B.
Quod erat demonstrandum
Incidentally, another, more descriptive, name for supremum is least upper bound.
Labels: mathematics