When I am trying to prove "Every bounded set in $R^n$ is totally bounded". I try to use a technique similar to the proof of "Every bounded set in $R$ is totally bounded". That is: I try to prove "For every $\epsilon$, every bounded set is covered in a finite number of cubes with diam $\leq \epsilon$". see this question
Since $A$ is bounded, let $A \subset \prod_{i=1}^n [a_i,b_i]$.
For each coordinate, we cut the real line as small intervals. That is, for each $i$ ($i=1,2, \cdots, n$), we let $(x_i)_0 = a_i$ and $(x_i)_j = a_i + j(\frac{b_i-a_i}{m})$. $m$ is big enough to make $\frac{b_i-a_i}{m} \leq \epsilon$
we can show $A \subset \prod_{i=1}^n [a_i,b_i] \subset \prod_{i=1}^n (\bigcup_{j=1}^m [(x_i)_{j-1},(x_i)_j])$.
Question 1: How to proceed? How to write the above formula as a union of cubes, by transforming the cartesian product of the union to the union of the cartesian product?
Question 2: Or is there a better way(I mean, elementary one) to solve this question?
Answer to question 2: In fact, a better solution is just this
