Originally Posted by

**Duke** let b>1. My textbook has asked me to do a prelimanary proof which I have answered

Define B(x) to be the set with elements b^t where t E Q and t<_ x, where x is real.

Show Sup(B(r))=b^r where r E Q

r>_t so b^r >_ b^t

Since Q is dense in R, for all y < r, there is a c E Q such that y<c<r, implying b^y<b^c<b^r so b^y is not an upperbound. Note all reals > 1 can be expressed b^y for some real y.

Hence b^x=Sup(B(x)) for real x.

So now to the proof in the title. b^x*b^y =Sup(B(x))Sup(B(y)) for real x,y

B(x+y)={b^t:t E Q, t<x+y}. I see I have to show Sup(B(x))Sup(B(y))=Sup(B(x+y))

but can't see how to proceed. Thanks