I want to show that for the dyadic intervals $I_{j,k}=[\frac{j}{2^k},\frac{j+1}{2^k})$, are either disjoint or $I_{n,m}\subset I_{j,k}$.
I was thinking of an argument that since the p-adic numbers equiped with the p-adic norm ($|\frac{rp^a}{s}|_p=p^{-a},\text{where p } \nmid \text{ r,s } \in \mathbb{N}, a \in \mathbb{N} )$ is an ultranorm and then use the properties of an ultrametric (the open balls are either disjoint or one is included in the other). I just have a feeling that it is a bit complicated. Is this a good way of showing it? Is there a way of doing it with just the "usual" absolute value on $\mathbb{R}$?
HINT: There’s no need to get fancy. Show that if $m=k$ and $n\ne j$, then $I_{n,m}\cap I_{j,k}=\varnothing$. Then assume without loss of generality that $m<k$. You’re now looking at the intervals
$$I_{n,m}=\left[\frac{n}{2^m},\frac{n+1}{2^m}\right)$$
and
$$I_{j,k}=\left[\frac{j}{2^k},\frac{j+1}{2^k}\right)\;,$$
where $m<k$. Let $\ell=k-m$; then
$$I_{n,m}=\left[\frac{n}{2^m},\frac{n+1}{2^m}\right)=\left[\frac{n2^\ell}{2^k},\frac{n2^\ell+2^\ell}{2^k}\right)\;.$$
Investigate what happens when $j<n2^\ell$, $n2^\ell\le j<n2^\ell+2^\ell$, and $n2^\ell+2^\ell\le j$.