Consider the following exercise: Let $(a,b)\in\mathbb{R}^n\times\mathbb{R}^m$ be an arbitrary point and let $\varepsilon>0$. Show that there are positive real numbers $\varepsilon_1,\varepsilon_2$ such that $$B^n(a,\varepsilon_1)\times B^m(b,\varepsilon_2)\subset B^{n+m}((a,b),\varepsilon)$$
This seems as an intuitive result, however, my problem is that I do not know how I should start writing this particular proof. Any tips, hint or help in general is welcome.