I am self-studying real analysis. I encountered the following assertion without a proof:
For each positive integer $k$, let $\mathcal{C}_k$ be the collection of all cubes of the form \begin{align*} \{(x_1,\dots,x_d):j_i2^{-k} \leq x_i < (j_i+1)2^{-k}\ \text{for}\ i = 1,\dots,d\}, \end{align*} where $j_1,\dots,j_d$ are arbitrary integers. Then
(1) each $\mathcal{C}_k$ is a countable partition of $\mathbb{R}^d$, and
(2) if $k_1<k_2$, then each cube in $\mathcal{C}_{k_2}$ is included in some cube in $\mathcal{C}_{k_1}$.
I tried a couple of examples, for instance, I tried $k=1$ and $j_i=i$, $k=1$ and $j_1=0$, $j_2=-5$, $j_3=10$, and so on. It does show that the claim is correct. However, I am really having a hard time proving it rigorously. I really appreciate it if someone could help me out!
I do not have the definition of a partition of $\mathbb{R}^d$. But I think it means a collection of nonempty disjoint sets whose union is $\mathbb{R}^d$.
Reference: Lemma 1.4.2 from Measure Theory by Donald Cohn
Frame challenge: this is not the kind of place in a real analysis course where putting in the effort required for a "formal proof" is a useful way to spend your time. The picture in the plane is good enough. Save your insistence on rigor for times that matter: the epsilons and deltas.
If you must fill in the details here I would start by showing that each cube in each of the partitions is a translate of a cube with "lower left corner" at the origin. That should get you the countability and the partition property (which you state correctly) in part (1) and the inclusion in part (2).