Question

Let A,B⊆R be nonempty. Define A−B={a−b|a∈A,b∈B}. Prove that inf(A−B)=inf(A)−sup(B)

Answer 1

Any ideas @Zarkon or @dirtydan667 ?

Answer 2

yes...I know how to prove it...I wanted to see if you have anything.

Answer 3

no im drawing a blank

Answer 4

I have nothing

Answer 5

can you prove...inf(A-B)=inf(A)+inf(-B)?

Answer 6

then show that inf(-B)=-sup(B)

Answer 7

both are standard results

Answer 8

I can prove inf(A+B)=inf(A)+inf(B)

Answer 9

Is the subtraction the same proof or no?

Answer 10

@Zarkon

Answer 11

You still there @Zarkon ?

Answer 12

no i killed him

Answer 13

@Zarkon can you please help me finish the proof?

Answer 14

if you can prove inf(A+B)=inf(A)+inf(B)

then That is the same as inf(A-B)=inf(A)+inf(-B)

let C=-B then

inf(A-B)=inf(A)+inf(-B)
is the same as

inf(A+C)=inf(A)+inf(C)

Answer 15

And then I show that inf(-b) is same as sup(b)

Answer 16

?