You shouldn't need the LUB property to show that 1/2^n → 0 as n→∞. It only requires the archimedean property. First prove (by induction) that 2^n>n. Then given ε>0, choose N with N>1/ε (that's where the archimedean property comes in). It will follow that (1/2)^n<ε whenever n≥N.