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6 Cards in this Set
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upper bound

if s is a subset of real numbers, then an upper bound for the set S is a number a is an element of R s.t. s is less than or equal for each s is an element S. If the set S has an upper bound we say that S is bounded from above.


We call a real number y a lub of the set S if

1. y is an upper bound for S
2. if a is any upper bound for S, then y is less or equal to a. 

if y is the lub and x is an element of R

x is less than y, then there exists an element s is an element s in set S s.t. x is less than s.


a nonempty finite subset S is a subset of real numbers always has a lub

in this case, the lub is simply the greatest element of S. Any subset S is a subset of real numbers that has a greatest element usu denoted max S has max S as a lub.


LUB property

A nonempty set of real numbers that is bounded from above has a lub


a is an element is a lower bound for the subset S of real numbers

if a is less or equal to s for each s is an element S, and a is a greatest lower bound of S if a is a lower bound of S and there exists no larger one. S is called bounded from below if it has a lower bound.
