Interchange of Function and Supremum
Theorem
Given a non decreasing function \(f\), and set of real numbers \(X\)
\[ f(\sup(X)) \geq \sup(f(X))\]
given that \(\sup(f(X))\) exists.
Proof
From the definition of the supremum
\[ \sup(X) \geq x \quad \forall x \in X\]
then, since \(f\) is non-decreasing
\[ f(\sup(X)) \geq f(x) \quad \forall x \in X.\]
Hence \(f(\sup(X))\) is an upper bound for \(f(X)\), and hence it is greater than or equal to the least upper bound, that is
\[ f(\sup(X)) \geq \sup(f(X)).\]