previous up next
Go backward to Illustration
Go up to Top
Go forward to Example
RISC-Linz logo

Unicity of Supremum


forall f, S0, S1:
   S0 is supremum of f /\  S1 is supremum of f => S0 = S1.
S is supremum of f : <=>
   S is upper bound of f /\ 
   (forall S': S' is upper bound of f => S <= S')
i.e., sup(f) = such S: S is supremum of f.

Proof: Take arbitrary f and suprema S0 and S1 of f. Since S0 is a supremum and S1 is an upper bound of f, we have S0 <= S1. Conversely, since S1 is a supremum and S0 is an upper bound of f, we have S1 <= S0. Since S0 <= S1 and S1 <= S0, we have S0 = S1.

Author: Wolfgang Schreiner
Last Modification: December 14, 1999

previous up next