I believe there is something wrong in my proof below because the result is not correct, can anyone help me spot that out?
$$ |f(x)|\geq f(x) \\ \text{So } |f(x)| \text{ is an upper bound of } f(x) \\ |f(x)|\geq \sup f(x) \\ \text{Also, } \sup |f(x)| \geq |f(x)| \\ \sup |f(x)|\geq\sup f(x)$$