Give an example of a function f for which [tex]\exists s \epsilon R P(s)[/tex] ^ Q(s) ^ U(s) P(s) is [tex]\forall x \epsilon R f(x) >= s[/tex] Q(s) is [tex]\forall t \epsilon R ( P(t) => s >= t )[/tex] U(s) is [tex]\exists y\epsilon R s.t. \forall x\epsilon R (f(x) = s => x = y) [/tex] So this was actually a two part question, and this is the second part, the first part involved the function f(x) = sin(x) for which P(s)^Q(s)^U(s) could never be true. I am not sure how to approach this, should I just think of random functions? or is there a logical way to do this. The only thing I know is what I found from the first part of this question which is P(s) defines the lower bound when true Q(s) defines the greatest lower bound when true U(s) I'm not so sure, I think something along the lines of y exist in reals, so the function must be all values of reals.. but this makes no sense as the function would not be bounded then. so I guess I am wrong. Any help would be greatly appreciated.