Suppose that a set S is bounded above. Let r be any real number and t be any positive real number.

Question

Let T = {r + tx : x e S}. Show that

sup T = r + t sup S

regards

Answer 1

Well, sup is a linear operator, so the normal rules of sum and product apply.

However, going back to the definition of sup,

x is the sup(S) if x >= s for s in S and there is no y<x such that y is also >=s for any s in S.

sup {r} is just r

sup tS is t sup S since if x >= s, then tx >= ts for any s in S.

Or something. Make it a bit more formal, and throw in addition, and it should hold.