# \$inf_Y[sup(2x+2y:0<x<1)]\$ How to approach this?

$$inf_Ysup_X f(x,y)$$
Let $$Y=(-1,0)$$ and $$X=(0,1)$$ and $$f(x,y)=2x+2y$$
Let’s start simply with $$inf_Y(sup(2x+2y:0
now here occurs the first problem for me, can I use the properties of $$Y$$ in this step, or would it be $$inf_Y(2+2y)$$ or since i know that Y is negative i could finish here and say $$inf_Y(2)=2$$
but the first approach would tell me it is: $$inf_Y(2+2y)=2-2=0$$ since $$Y=(-1,0) =>inf_Y(2y)=-2$$

So do I have to take sup of f(x,y) regardless of what Y is, which doesn’t make sense. Maybe i need a proper definition for multivaraible sup and inf

Please give a comprehensive answer which approach is correct. Any explanation is appreciated