I have another question. Would someone mind showing me how to do this? Let S be a bounded nonempty subset of the real numbers. Let a and b be fixed real numbers. Define the set T as {as+b: s in S}. Find formulae for supT and infT in terms of supS and supT and prove your formulae.
Thank you for any help and thanks for help from before.
How many different combinations are there? I'm thinking I only need to find one formula for each of the supT and infT. Yet, it appears the formulae are different depending on whether a is positive or negative, etc. I'll probably be able to prove whatever the answer is, but can you just lead me in the right direction to actually get the formula?
It seems for a>0, supT= asupS+b and infT= ainfS+b. For a<0, supT= ainfS+b and infT=asupS+b.
In an analysis type course, do you think you need to use epsilon notation to prove these?
Is it ok just to say that supS is a supremum and multiplying it by a constant a and adding a constant b (the definition of the set T) will make this the supremum also?