Compact sets are bounded: shape of the cover matters? To prove a compact sets is bounded, we assume there's a "open ball cover" (each with R=1) that covers the set. And take maximum distance over the center of the balls +2 as the boundary. Why could we assume this without actually knowing the "shape" of the cover? How do we know if such "open ball cover" is able to cover the set presumably? Maybe such "ball-shaped" cover is not able to cover the set.

The usual strategy is to let the open cover of the compact set $K$ be $\mathcal B=\{B(x,1):x\in K\}$. Since each point in $K$ has its own ball, $\mathcal B$ definitely covers $K$. Since $K$ is compact, there exists a finite subcover.
