How to show $d>0$?

Question

Suppose $A$ is a non-empty bounded closed set in $\Bbb C$, and $A$ does't have accumulation points in $\Bbb C$.

We denote $d=\inf |a-b|, a,b\in A, a\neq b$.

How to show $d>0$?

Answer 1

Since $A$ is compact and does not have accumulation points in $\mathbb C$, $A$ must be finite !

Proof: suppose that $A$ is infinite. Then there is a sequence $(a_n)$ in $A$ such that $a_n \ne a_m$ for $n \ne m.$.

$A$ is compact, hence $(a_n)$ contains a convergent subsequence with limit $a \in A.$

Then $a$ is an accumulation point of $A$, a contradiction.

Answer 2

Since $A$ is a bounded closed set, it is compact. $\forall a\in A$, we can choose an open set $U_a\ni a$ such that $U_a\cap (A-a)=\varnothing$, because $A$ is compact, $\{U_a\}_{a\in A}$ is a finite set, so $A$ is a finite set.