Real Analysis: intersection interval proof?? Please help!?

Question

Real Analysis:

Prove the intrsection interval from n=1 to infinity (0,1/n)=empty. (cant really do the intersection notation)... Notice that this demonstrates that the intervals in the nested interval property must be closed for the conclusion of theorem to hold.

Thanks!!

Answer 1

Suppose x∈[n=1, ∞]⋂(0, 1/n). Then clearly x∈(0, 1) and so in particular x>0. Thus by the archimedian property, ∃k s.t. x>1/k. But then x∉(0, 1/k), and so a fortiori, x∉[n=1, ∞]⋂(0, 1/n) -- a contradiction. Therefore [n=1, ∞]⋂(0, 1/n) has no elements, and is equal to the empty set. Q.E.D.

Answer 2

Suppose it weren't empty. Let x be in the intersection. x>0, since every number in every one of those intervals is >0.

Pick N > 1/x. Then x >1/N. So x isn't contained in (0,1/N). Contradiction. QED.