Problem 5 Study Guide

Prove that $\inf({\{\frac{1}{\sqrt{n}}:n\in\mathbb{N}}\})=0$
Proof:
We must prove two things:

1. That zero is a lower bound, and
2. that any number larger than zero is no longer an upper bound.

So,

1. 0 is a lower bound for the set, since $\frac{1}{\sqrt{n}}>0$ for all $n$.
2. Next, suppose that $a>0$ and choose $n\in\mathbb{N}$ such that $n>\frac{1}{a^2}$. Then, $\frac{1}{\sqrt{n}}< a$. Thus, $a$ is not a lower bound.

Personally, I would like to see that the Archimedean property, explicitly stated in the proof, is the means by which we have the ability to choose $n$ in Step 2 because that isn't trivial.

You didn't exactly conclude anything at the end. "Thus, $a$ is not a lower bound." Does this mean that 0 is? Could it be: Thus, $a$ is not a lower bound and $0$ is the greatest lower bound. @agibson

I like it! I did expand it just a bit, as it was a bit terse, but the main ideas were there.