Im stuck with this exercise
For $A\subset\Bbb R^n$ and $B\subset \Bbb R^m$ show that $\dim_H(A\times B)=\dim_H(A)+\dim_H(B)$
where $\dim_H$ is the Hausdorff dimension. I know that when $A$ and $B$ are open the above holds. However Im unable to generalize the result.
I tried to relate the following inequalities and identities
$$\operatorname{diam}(A)\lor\operatorname{diam}(B)\le\operatorname{diam}(A\times B)\le\operatorname{diam}(A)+\operatorname{diam}(B)\tag1$$
$$\mathcal H_*^r(A)<\infty\implies\mathcal H_*^s(A)=0,\quad\forall s>r\\ \mathcal H_*^r(A)>0\implies\mathcal H_*^s(A)=\infty,\quad\forall s\in[0,r)\tag2$$
$$x,y\in[0,1]\implies xy<x+y\,\text{ and }\, x^{r+s}<x^r,\quad\forall r,s>0\tag3$$
$$\inf A+\inf B=\inf(A+B)\text{ and }\sup A+\sup B=\sup(A+B)\tag4$$
where $\mathcal H_*^s$ is the $s$-dimensional Hausdorff outer measure and $\rm diam$ is the diameter of a set. By example I find that
$$\dim_H(A)+\dim_H(B)=\inf\{r>0:\exists \alpha\in[0,r]\text{ such that }\mathcal H_*^{r-\alpha}(A)+\mathcal H_*^\alpha(B)=0\}\tag5$$
so a line of action is try to relate $(2)$ and $(5)$ in something like
$$\mathcal H_*^{r-\alpha}(A)+\mathcal H_*^\alpha(B)=0\implies\mathcal H_*^r(A\times B)=0\tag6$$
using the definition of $\mathcal H_*^s$, $(1)$ and maybe $(3)$. However I found nothing. Some help will be appreciated.
EDIT: to clarify some things: from the definitions of Hausdorff dimension the statement to be proved can be stated as $$ \begin{align}\dim_H(A)+\dim_H(B)&=\inf\left\{s+t>0:\sup_\epsilon\inf\left\{\sum_{k=0}^\infty a_k^r+b_k^s: a_k,b_k<\epsilon\right\}=0\right\}\\ &=\inf\left\{s+t>0:\sup_\epsilon\inf\left\{\sum_{k=0}^\infty c_k^{r+s}: c_k<\epsilon\right\}=0\right\}\\ &=\sup\left\{s+t\ge0:\sup_\epsilon\inf\left\{\sum_{k=0}^\infty a_k^r+b_k^s: a_k,b_k<\epsilon\right\}=\infty\right\}\\ &=\sup\left\{s+t\ge0:\sup_\epsilon\inf\left\{\sum_{k=0}^\infty c_k^{r+s}: c_k<\epsilon\right\}=\infty\right\}\\ &=\dim_H(A\times B)\end{align}\tag{*} $$ where $a_k,b_k,c_k$ are the diameters of sequences of covers $(A_k),(B_k),(C_k)$ of $A\subset\Bbb R^n$, $B\subset\Bbb R^m$ and $A\times B\subset\Bbb R^{n+m}$ respectively.
This together with $(2)$ seems the way to go, however I can't found appropriate bounds because I can't relate covers $A$ and $B$ with covers of $A\times B$ such that it make possible to find these bounds.
At my disposal, in the context where this exercise appear, there is not too many theorems to solve this exercise, by example I dont know the Frostman's lemma that @DavidUlrich state in the comment. I need to solve it from elementary theorems as the stated above.
P.S.: I know some more identities related to the Hausdorff dimension, by example that is an increasing function or that $\dim_H(f(A))\le\dim_H(A)$ for $f$ Lipschitz or that $\dim_H(\bigcup_k A_k)=\sup_k\dim_H(A_k)$.