real analysis - Must an unbounded set in a metric space be infinite?

I'm new to real analysis and topology. Recently, I'm reading baby rudin. Occasionally, I've a question: does a set is unbounded implies the set is infinite in metric space? I think the statement is right, but I can't prove it.

Please give the strict proof. Thanks in advance.

Answer

Yes (if metric spaces are assumed to be non-empty).

Let $\langle X,d\rangle$ denote a metric space.

Suppose that a set $S\subseteq X$ is finite and let $x\in X$.

If we take $r>\max\{d(x,y)\mid y\in S\}$ then $S\subseteq B(x,r)=\{y\in X\mid d(x,y)

That means that unbounded sets cannot be finite, hence are infinite.

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Note: this answer preassumes that metric spaces are not empty. If $X=\varnothing$ then the finite set $\varnothing$ is unbounded since $X=\varnothing$ is not contained in any ball centered at some $x\in X$. This because balls like that simply do not exist in that situation.