Are sets everywhere dense in rational numbers set?

Question

I'm new in MSE and in functional analysis. Really need explanation with this topic and exercise: Is the set {${m^2}/{n^2}: m,n\in \mathbb{N}$} everywhere dense in positive rational numbers set? And is the set {${m^m}/{n^n}: m,n\in \mathbb{N}$} everywhere dense in positive rational numbers set? I thought to use here a dense set definition but maybe it isn't enough.

Answer 1

I'm going to assume you mean the positive rationals, since otherwise the answer is trivially "no" (do you see why?).

Here are two hints:

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Note that the first set is just the set of positive rational squares $\{q^2: q\in\mathbb{Q}\}$ (why?). Now suppose I have a $p\in\mathbb{Q}_{>0}$, and I want to find a positive rational square "really close" to $p$.

• Although $\sqrt{p}$ probably isn't rational (e.g. take $p=2$), there are positive rationals really really close to $\sqrt{p}$ (why?).

• Intuitively, if $q$ is a positive rational really really close to $\sqrt{p}$, then $q^2$ should be really close to $p$. Do you see how to make this precise, and why this implies that the set of positive rational squares is dense in $\mathbb{Q}_{>0}$?

Before we move on to the next part, here's a follow-up exercise: what about the set of positive rational cubes? Is it dense in $\mathbb{Q}_{>0}$?

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The above trick doesn't seem to work - at least, not immediately - for the second set, so it's time to play around a bit! Testing some possible choices of $m$ and $n$ (actually do this) should convince you that numbers of the form ${m^m\over n^n}$ are either "really big" or "really small." That kind of sparseness suggests that the set isn't dense. To start working your way towards a proof, suppose I have natural numbers $m<n$. What's the biggest possible value for ${m^m\over n^n}$? It's certainly at most $1$, but can you find a better bound?

• Before tackling this immediately, try some examples - $m=1,n=2$, and $m=1,n=3$, and $m=2,n=3$, and maybe a couple more until you can make a guess.

• Once you've proved that your guess is correct: what does this bound tell you about the density/non-density of the set $\{{m^m\over n^n}:m,n\in\mathbb{N}\}$? HINT: think about what happens between this bound and $1$ ...