The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.
An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now we divide the square $S$ into four smaller squares by joining the pairs of midpoints of opposite sides. One of these square will not have a finite subcover from the given cover. For, otherwise, each of these four squares will have finite subcover so that the union of these subcovers will be a cover for $S$. Thus, $S$ itself will admit a finite subcover.
We then choose one such smaller square and call it $S_1$ . And then repeat the process by subdividing $S_1$ into four squares and choosing one of the smaller squares which does not admit a finite subcover.
In this way we get a sequence of squares $S_n$ such that $S$, dose not admit a finite subcover and the length of sides of $S_n$ gradually decreases.
I think we will get a contradiction from here. Please Help to write a formal proof.