If S = {1/n | n ∈ ℕ}, what is inf(S)?

Question

If $S = \{1/n \mid n ∈ ℕ\}$, what is $\inf(S)$?

I believe the answer is $0$, but I'm not really sure how to prove it...does it involve using epsilon?

Answer 1

Obviously the set is bounded below by 0, so $\inf(S)\geq 0$. Assume $\inf(S):=\varepsilon > 0$. Then by the Archimedean property of the naturals (i.e. since the naturals are not bounded above), we can find a natural $N$ s.t. $1/N < \varepsilon$, but then $\inf(S) \leq \inf(\{1/N\})<\inf(S)$, a contradiction.