Finding sup{1/n : n ∈ ℕ}

Real Analysis · Axiom Academy

A rigorous worked example: the supremum is attained (a maximum), while the infimum is not. Let . Find , and decide whether it is a maximum. As a contrast, find and decide whether it is a minimum. The terms 1, 1/2, 1/3, 1/4, … march left and crowd toward 0 without ever reaching it. The largest term, 1, sits at the right end and belongs to the set. Nice work. You found and for , and you saw why one is attained and the other is not. Sup that is attained = maximum: is an upper bound that belongs to S , so . Inf that is not attained: 0 is the greatest lower bound, but since for every n — so S has no minimum. The Archimedean property does the work: for any there is an n with (take ), so no number above 0 can be a lower bound. Sup/inf need not be in the set: " " and " " always exist for a bounded set of reals, but only become " " / " " when they are actually achieved. The pattern generalizes: to show a candidate bound is the supremum, prove it bounds the set and that nothing smaller does — and check separately whether it is reached.

This is the written version of the interactive lesson above. See the full Real Analysis course.