A fact about R which is surprisingly not a consequence of P1-P12 is the Archimedian Property:
The set of natural numbers is not bounded above.
Or in other words: for every R belongs to R, there is an integer N belongs to N so that N>R. Assume that the Archimedian Property holds in the questions below.
(a) Prove that for every epsilon>0 there exists a natural number N so that 1/N<epsilon(hint: assume false , take reciprocals, compare with the Archimedian Property)
(B ) suppose the rational number r belongs to the interval(a-delta,a+delta).show that this interval also contains an irrational number.(hints:how large do you have to make N so that r+ sqrt2/N is guaranteed to belong to (a-delta, a+delta)?justify the existence of N, why is r+sqrt2/N irrational?)
©show that for every delta and every a, the interval(a-delta, a+delta)contains both rational numbers and irrational numbers.
how to do b and c?