Let $((a_i,b_i))$ $\in$ $(]0,1[²)^\mathbb N $ such that
$\sum_{i\in\mathbb N}$$a_i$$b_i$ = $+\infty$
Show that all compact set can be covered by translates of cobbles $[0,a_i]$ $×$ $[0,b_i]$.
In my lesson, a compact is defined as a set in which all sequences possess a sub-sequence that converges toward a limit that is in the set.
Thank you.