I am trying to show that an "epsilon ball in the taxicab metric" can be contained in a "epsilon ball in the usual metric". Let "Up" be a taxicab ball, and "Ud" be a usual metric ball. How can I show that Up( (x,y) , e ) is a subset of Ud( (x,y), e ) ? I have drawn the picture and I understand how the taxicab ball is inside. It is a diamond that touches the usual ball on each of the diamond's four corners and is less than everywhere else. How can I show this algebraically?