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. 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.
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