Subgroups of (R, +) are Dense or Cyclic

I prove all subgroups of additive group of real numbers are either cyclic or dense. Then, we use that to show the set of rationals of the form 2^m 3^n is dense in [0, infty).

[calls]

Thank you for this amazing explanation, I was struggling to understand this.

[zeggwaghismail]

Very well done. Density is on trend everywhere now!

[derkritiker]

I think in 9:30 the ineaquality should be a<b<a+ε right?

[ilemt]

I'm interested in this. Could you provide some references or materials about this topic?

[acrommclain]

I have a question about a slight generalization.

Let $A = \{(2^{n}3^{m}, 5^{n}7^{m}):\space n, m\in\mathbb{Z}\}\subset\mathbb{R}^{2}$

What is $D(A)$ (the set of the accumulation points of A)?

Is A dense in $\mathbb{R}^{2}?

What if $A = \{(2^{n}3^{m}, 5^{n}7^{m}):\space n, m\in\mathbb{Z}\spacen\leq0\leqm\}\subset\mathbb{R}^{2}$?

[jiaweihuo]

Does it mean that for any different numbers $a$ and $b$ of $\mathbb{R}$, then the additive group of $a, b$ is dense in $\mathhb R$?

[srishtinegi]

So If it is dense it must not be cyclic, right?

[adfr]

I dont understand what happens at 10:40, you just showed the infimum belongs to G not that there is an elements smaller than epsilon... what am I missing