Uniformly continuous function on a totally bounded set is bounded
= Uniformly continuous function on a totally bounded set is bounded
Choose the <uniform continuity> input tolerance corresponding to output tolerance one. A <totally bounded> domain is covered by finitely many balls of smaller radius centred in the domain. Every function value differs by less than one from the value at one of their finitely many centres. Taking the largest centre value bounds the whole function. Compactness or closedness of the domain is unnecessary.