Finite subset dense in a compact space

Question

If $M$ is compact and $A$ is dense in $M$, prove that for any $\delta > 0$ there is a finite subset $\{a_1, \dots, a_k\} \subset A$ which is $\delta$-dense in $M$ in the sense that each $x \in M$ lies within distance $\delta$ of the points $a_j$, $j = 1, \dots, k$.

Answer 1

If $\;B(y,r)\;$ denotes the open ball with center $\;y\;$ and radius $\;r>0\;$ , then $\;\left\{\,B(a,\delta)\,\right\}_{a\in A}\;$ is an open cover of $\;M\;$ (why?), and since $\;M\;$ is compact there exists a finite subcover $\;\left\{\,B(a_i,\delta)_{1\le i\le k}\,\right\}\;$ ... finish the argument.