Is a waiver of a defendant's right to a jury trial valid?

At the beginning of his trial, Mills told the court that he had signed a waiver of his right to a jury trial. When questioned he said that he had reviewed the matter with his attorney, knew he had a right to a jury trial, wanted to waive that right, and desired to be tried by the court. Mills now argues that despite his admitted execution of the written waiver, the trial court's examination was insufficient to render his waiver knowing and voluntary. A valid waiver of a defendant's right to a jury trial must be "made voluntarily, knowingly and intelligently." United States v. Cochran, 770 F.2d 850, 851 (9th Cir.1985).