# Thread: infimum and supremum proof

1. ## infimum and supremum proof

Prove, if S is a nonempty subset of real numbers that is bounded below then
inf(S)= -sup{-s: s in S}.

I have shown that the {-s:s in S} is bounded below and I let u = sup of that set but I do not know how to show -u = inf(S).

2. Suppose that $\displaystyle \lambda = \inf (S)$ then if follows that $\displaystyle \left( {\forall z \in S} \right)\left[ {\lambda \le z \Rightarrow \quad - z \le - \lambda } \right]$.
That mean that $\displaystyle -\lambda$ is an upper bound of the set $\displaystyle -S$.
Say that $\displaystyle \kappa < - \lambda$ then $\displaystyle \lambda < - \kappa$.
By the definition of infimum $\displaystyle \left( {\exists y \in S} \right)\left[ {\lambda \le y < - \kappa } \right]$
That means that $\displaystyle \left[ {\left( { - y \in - S} \right) \wedge \left( {\kappa < - y \le - \lambda } \right)} \right]$.
This means that no number less that $\displaystyle -\lambda$ is an upper bound of $\displaystyle -S$.
Therefore $\displaystyle - \lambda = - \inf (S) = \sup ( - S)$.