prove $\inf \frac{1}{1+n}$ is zero

Question

${a_n}={\frac{1}{1+n}}$ for all natural numbers.
how to prove $inf(a_n)=0$
I need help with the part that I have to apply the archimedian axiom. should I say there exists an N such that $\frac{1}{1+N}<0+\epsilon$
should I begin with $N>\frac{1}{\epsilon}$ then how to develop that. I need a clean proof for this part please. Thanks

Answer 1

Yes. From $N>\frac1\varepsilon$, you deduce that $1+N>\frac1\varepsilon$ and that therefore $\frac1{1+N}<\varepsilon$. So, no $\varepsilon>0$ is a lower bound of your set. It follows that $0$ is the greatest lower bound.